So is the “input” to this computation the functions U and P? Is “that computation” all places in spacetime when this particular input was considered, or all uses of the TDT framework at all?
“This computation” is exactly equal to the Godelian diagonal and anything you can deduce from making assumptions about it. If I assume the output of a calculator into which I punched “3 + 3″ is “6”, then the question is not “What computation do I believe this to be, exactly?” but just “What else can I logically infer from this given my belief about how various other logical facts are connected to this logical fact?” You could regard the calculator as being a dozen different calculations simultaneously, and if your inferences are sound they ought not to tangle up.
With that said, yes, you could view the TDT formula as being parameterized around U, P, and the action set A relative to P. But it shouldn’t matter how you view it, any more than it matters how you view a calculator for purposes of making inferences about arithmetic and hence other calculators. The key inferences are not carried out through a reference class of computations which are all assumed to be correlated with each other and not anything else. The key inferences are carried out through more general reasoning about logical facts, such as one might use to decide that the Taniyama Conjecture implied Fermat’s Last Theorem. In other words, I can make inferences about other computations without seeing them as “the same computation” by virtue of general mathematical reasoning.
“That computation” is just a pure abstract mathematical fact about the maximum of a certain formula.
Counterexample request: can you give me a specific case where it matters which computation I view myself as, given that I’m allowed to make general mathematical inferences?
I really have a lot of trouble figuring out what you are talking about. I thought I could take just one concept you referred to and discuss that, but apparently this one concept is in your mind deeply intertwined with all your other concepts, leaving me without much ground to stand on to figure out what you mean. I guess I’ll just have to wait until you write up your ideas in a way presentable to a wider audience.
I agree that if we had a general theory of logical uncertainty, then we wouldn’t need to have an answer to Robin’s question.
Counterexample request: can you give me a specific case where it matters which computation I view myself as, given that I’m allowed to make general mathematical inferences?
I think the old True PD example works here. Should I view myself as controlling the computation of both players, or just player A, assuming the two players are not running completely identical computations (i.e. same program and data)? If I knew how I should infer the decision of my opponent given my decision, then I wouldn’t need to answer this question.
What I would generally say at this point is, “What part of this is a special problem to TDT? Why wouldn’t you be faced with just the same problem if you were watching two other agents in the True PD, with some particular partial knowledges of their source code, and I told you that one of the agents’ computations had a particular output? You would still need to decide what to infer about the other. So it’s not TDT’s problem, it legitimately modularizes off into a magical logical inference module...”
(Of course there are problems that are special to TDT, like logical move ordering, how not to infer “A1 has EU of 400, therefore if I output A2 it must have EU > 400”, etc. But “Which computation should I view myself as running?” is not a special problem; you could ask it about any calculator, and if the inference mechanism is sound, “You can use multiple valid abstractions at the same time” is a legitimate answer.)
So is the “input” to this computation the functions U and P? Is “that computation” all places in spacetime when this particular input was considered, or all uses of the TDT framework at all?
“This computation” is exactly equal to the Godelian diagonal and anything you can deduce from making assumptions about it. If I assume the output of a calculator into which I punched “3 + 3″ is “6”, then the question is not “What computation do I believe this to be, exactly?” but just “What else can I logically infer from this given my belief about how various other logical facts are connected to this logical fact?” You could regard the calculator as being a dozen different calculations simultaneously, and if your inferences are sound they ought not to tangle up.
With that said, yes, you could view the TDT formula as being parameterized around U, P, and the action set A relative to P. But it shouldn’t matter how you view it, any more than it matters how you view a calculator for purposes of making inferences about arithmetic and hence other calculators. The key inferences are not carried out through a reference class of computations which are all assumed to be correlated with each other and not anything else. The key inferences are carried out through more general reasoning about logical facts, such as one might use to decide that the Taniyama Conjecture implied Fermat’s Last Theorem. In other words, I can make inferences about other computations without seeing them as “the same computation” by virtue of general mathematical reasoning.
“That computation” is just a pure abstract mathematical fact about the maximum of a certain formula.
Counterexample request: can you give me a specific case where it matters which computation I view myself as, given that I’m allowed to make general mathematical inferences?
I really have a lot of trouble figuring out what you are talking about. I thought I could take just one concept you referred to and discuss that, but apparently this one concept is in your mind deeply intertwined with all your other concepts, leaving me without much ground to stand on to figure out what you mean. I guess I’ll just have to wait until you write up your ideas in a way presentable to a wider audience.
I agree that if we had a general theory of logical uncertainty, then we wouldn’t need to have an answer to Robin’s question.
I think the old True PD example works here. Should I view myself as controlling the computation of both players, or just player A, assuming the two players are not running completely identical computations (i.e. same program and data)? If I knew how I should infer the decision of my opponent given my decision, then I wouldn’t need to answer this question.
What I would generally say at this point is, “What part of this is a special problem to TDT? Why wouldn’t you be faced with just the same problem if you were watching two other agents in the True PD, with some particular partial knowledges of their source code, and I told you that one of the agents’ computations had a particular output? You would still need to decide what to infer about the other. So it’s not TDT’s problem, it legitimately modularizes off into a magical logical inference module...”
(Of course there are problems that are special to TDT, like logical move ordering, how not to infer “A1 has EU of 400, therefore if I output A2 it must have EU > 400”, etc. But “Which computation should I view myself as running?” is not a special problem; you could ask it about any calculator, and if the inference mechanism is sound, “You can use multiple valid abstractions at the same time” is a legitimate answer.)