I can certainly imagine a universe where none of these concepts would be useful in predicting anything, and so they would never evolve in the “mind” of whatever entity inhabits it.
Can you actually imagine or describe one? I intellectually can accept that they might exist, but I don’t know that my mind is capable of imagining a universe which could not be simulated on a Turing Machine.
The way that I define Tegmark’s Ultimate Ensemble is as the set of all worlds that can be simulated by a Turing Machine. Is it possible to imagine in any concrete way a universe which doesn’t fall under that definition? Is there an even more Ultimate Ensemble that we can’t conceive of because we’re creatures of a Turing universe?
The set of all Turing Machines is merely countable. Instead, imagine an ensemble of universes corresponding to the real numbers, some of which aren’t even computable.
For example, universes running on classical mechanics, where values can be measured with infinite precision, and which also have continuous space and time to represent those infinitely precise values. In other words, a set of universes where two universes can be different in the position of a particle, position being defined as a real number.
This seems easy to imagine—it’s a pretty standard Newtonian world.
I can’t either. But I can imagine a partially predictable world, which cannot be perfectly simulated by any Turing machine (that’s what Gavin’s question was about), and yet is completely deterministic and has a simple mathematical description. For instance, a continuous, deterministic world where a fundamental physical constant is an uncomputable number.
I see—you’re going after a much weaker criterion than the one described in the OP.
I was responding to Gavin’s comment and its criterion was “a universe which could not be simulated on a Turing Machine”.
I’d note that my ‘finite precision’ comment covers uncomputable constants. Unless no computation can even converge on them...
The set of all constants on which some computation converges is still only countable (each Turing machine’s output converges on at most one constant).
A universe with uncomputable constants in the laws of nature might still be predictable in practice. We don’t know if the constants in our own universe are computable or not, because we can’t even measure them beyond a few hundreds bits each. And as long as we keep measuring them instead of deriving them from some equation, our knowledge will presumably remain finite, so computability does not matter for us in practice.
On the other hand, full simulation does depend on the precise values. (If it didn’t, the constants wouldn’t have those precise values in any meaningful sense.)
That seems to be the same as ‘no computation can output a prefix that is sufficiently long to be useful in predictions’?
This would depend on how long a prefix you need. If there exists a constant K such that K-length prefixes of all universal constants are enough, then certainly all possible bitstrings of length K can be computed.
The question becomes: what does a universe look like where you need to know the exact value of some constants to make useful predictions? (Perhaps you can make some predictions using approximations, but some other possible scenarios remain unpredictable without precise knowledge.)
This requires knowing an infinite amount of information: some kind of physical reification of an uncomputable real number. (I’m only using real numbers as a familiar example; larger infinities could be used too.) Not just to know the constant, but probably also to know the precise initial state of the system whose evolution you want to predict.
One possible reification is the precise position, mass, momentum, etc. of an element in the simulation—assuming these properties can also have any real value, and that you can control them to that degree.
I feel I could still imagine such a universe. But either way that’s a fact about me, not about universes. I don’t know how to predict the measure of such universes in any multiverse theory.
I would think those would all be representable by a Turing Machine, but I could be wrong about that. Certainly, my understanding of the Ultimate Ensemble is that it would include universes that are continuous or include irrational numbers, etc.
I don’t know that my mind is capable of imagining a universe which could not be simulated on a Turing Machine.
I never said it could not be, just that the Turing Machine would not be a concept that is likely to evolve there.
Imagine a universe where there are no discrete entities, so numbers/addition is not a useful model. Whatever inhabits such a universe, if anything, would not develop the abstraction of counting. This universe could still be Turing-simulated (Turing Machine is an abstraction from our universe),
This is the essential point I am trying to make. Mathematics is determined by the structure of the universe and is not an independent abstract entity. I feel like I failed, though.
Can you actually imagine or describe one? I intellectually can accept that they might exist, but I don’t know that my mind is capable of imagining a universe which could not be simulated on a Turing Machine.
The way that I define Tegmark’s Ultimate Ensemble is as the set of all worlds that can be simulated by a Turing Machine. Is it possible to imagine in any concrete way a universe which doesn’t fall under that definition? Is there an even more Ultimate Ensemble that we can’t conceive of because we’re creatures of a Turing universe?
The set of all Turing Machines is merely countable. Instead, imagine an ensemble of universes corresponding to the real numbers, some of which aren’t even computable.
For example, universes running on classical mechanics, where values can be measured with infinite precision, and which also have continuous space and time to represent those infinitely precise values. In other words, a set of universes where two universes can be different in the position of a particle, position being defined as a real number.
This seems easy to imagine—it’s a pretty standard Newtonian world.
Even in a Newton world, you can make predictions. You just won’t be able to make them to infinite precision.
I, too, am having trouble imagining a perfectly unpredictable universe with agents in it.
I can’t either. But I can imagine a partially predictable world, which cannot be perfectly simulated by any Turing machine (that’s what Gavin’s question was about), and yet is completely deterministic and has a simple mathematical description. For instance, a continuous, deterministic world where a fundamental physical constant is an uncomputable number.
I see—you’re going after a much weaker criterion than the one described in the OP.
I’d note that my ‘finite precision’ comment covers uncomputable constants. Unless no computation can even converge on them...
I was responding to Gavin’s comment and its criterion was “a universe which could not be simulated on a Turing Machine”.
The set of all constants on which some computation converges is still only countable (each Turing machine’s output converges on at most one constant).
A universe with uncomputable constants in the laws of nature might still be predictable in practice. We don’t know if the constants in our own universe are computable or not, because we can’t even measure them beyond a few hundreds bits each. And as long as we keep measuring them instead of deriving them from some equation, our knowledge will presumably remain finite, so computability does not matter for us in practice.
On the other hand, full simulation does depend on the precise values. (If it didn’t, the constants wouldn’t have those precise values in any meaningful sense.)
I meant to say ‘no computation can even constrain them to a useful degree’. Strict convergence is not necessary.
That seems to be the same as ‘no computation can output a prefix that is sufficiently long to be useful in predictions’?
This would depend on how long a prefix you need. If there exists a constant K such that K-length prefixes of all universal constants are enough, then certainly all possible bitstrings of length K can be computed.
The question becomes: what does a universe look like where you need to know the exact value of some constants to make useful predictions? (Perhaps you can make some predictions using approximations, but some other possible scenarios remain unpredictable without precise knowledge.)
This requires knowing an infinite amount of information: some kind of physical reification of an uncomputable real number. (I’m only using real numbers as a familiar example; larger infinities could be used too.) Not just to know the constant, but probably also to know the precise initial state of the system whose evolution you want to predict.
One possible reification is the precise position, mass, momentum, etc. of an element in the simulation—assuming these properties can also have any real value, and that you can control them to that degree.
I feel I could still imagine such a universe. But either way that’s a fact about me, not about universes. I don’t know how to predict the measure of such universes in any multiverse theory.
I would think those would all be representable by a Turing Machine, but I could be wrong about that. Certainly, my understanding of the Ultimate Ensemble is that it would include universes that are continuous or include irrational numbers, etc.
Turing Machines have discrete, not continous states. There is a countable infinity of Turing Machines.
I never said it could not be, just that the Turing Machine would not be a concept that is likely to evolve there.
Imagine a universe where there are no discrete entities, so numbers/addition is not a useful model. Whatever inhabits such a universe, if anything, would not develop the abstraction of counting. This universe could still be Turing-simulated (Turing Machine is an abstraction from our universe),
This is the essential point I am trying to make. Mathematics is determined by the structure of the universe and is not an independent abstract entity. I feel like I failed, though.