(Note you couldn’t have too many facts, or they’d self-contradict.)
I can remove contradictions by making up a new rule that specifies which rules take precedence over other ones.
Let’s build a simpler intuition pump: instead of universes, we’ll talk about infinite sequences of integers that can be specified by finite sets of rules. For example, (1, 1, 1, …) is such a sequence. (1, 2, 3, 4, 5, …) is another. Those look regular all right. But this sequence: (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) can also be specified by a finite and self-consistent set of rules, even though something seems to have “swooped down” and changed it in one place. There’s no hard difference between “regular-looking” and “irregular-looking” sequences. All finite sets of rules have equal footing.
Does this make sense to you? Now imagine those integers encoding the time evolution of your toy universe...
It seems to me we’re conflating ‘possible as an output’ and ‘true’ but since I don’t really know what ‘true’ means in this context, let’s conflate them.
The fact that you’ve written the sequence (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) is evidence that it’s the possible output of an algorithm. (Indeed, it was the output of an algorithm you ran.) However, this means that the output was possible (and true) for a subset of the universe. How do you know this rule could be universally true?
I say that it could not be universally true, because it has this property of arbitrariness. I think to answer, ‘what could possibly be universally true?’, you would have to answer the question, ‘what can be deduced as true from nothing?’ or at best, ‘what can be deduced as possibly true from nothing?’ From nothing, the universe might deduce the natural numbers. By definition of what the natural numbers are they have an ordering 1, 2, 3, …, n, n+1, … This ordering really could not be different.
Suppose that the universe had a way of “knowing about” a single element ’12345′ that it places in a new position between 1000000 and 1000001. Simultaneously, it could have placed this element in any position, so universally, you would get a much, much larger structure in which (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) was only a tiny subset.
My point, whether I can figure out how to make it or not, is that the universe doesn’t ‘know about’ 12345, it only knows about all the numbers and evolves this structure universally. You can look at particular components of the structure and observe that individual components seem arbitrary, but the entire structure cannot be.
Imagine a “universe” that consists of all streams of natural numbers that can be specified by algorithms. Is that fundamental and non-arbitrary enough for you? This universe contains many “sub-universes” that cannot communicate, so we can call them “universes” in their own right. One of them is my 12345 sequence, and many others have me spontaneously turn into a pheasant a week from now.
Imagine a “universe” that consists of all streams of natural numbers that can be specified by algorithms. Is that fundamental and non-arbitrary enough for you?
Exactly, yes.
This universe contains many “sub-universes” that cannot communicate, so we can call them “universes” in their own right.
How do you know they don’t communicate? This would be a very non-trivial claim. I’m saying that the set of things that could be independently true (and thus universally true) might be extremely small, and certainly much smaller than the set of possibly-possible things you can think of. Most things we can think of as possible are going to be entangled in ways we aren’t aware of with other truths.
Instead of being where you are thinking of things that could be (‘I turn into a pheasant in 5 minutes’), you would need to turn it upside down and think if there was nothing, what would be true? Perhaps not so many things … perhaps this experienced universe is the only one that was possible. How do we know without developing a theory about what truths self-generate from a void?
Same way I know natural numbers don’t communicate. The output of one algorithm can’t “communicate” with the output of another algorithm, whatever that means.
Each possible universe corresponds to a different set of axioms, right? (If two universes have exactly the same axioms, then they’ll be the same, and the addition of any new axiom that is consistent with but not deducible from the others will make a new universe.)
I’ve maintained all along that arbitrary and weird sets of rules can occur in subsets of the universe, but should not be universally true (true throughout a universe) because there cannot exist a set of axioms that would result in these rules. For example, you can build possibly a machine that turns someone into a pheasant and an extra bag of sand, even in this universe, but it wouldn’t ever be a universal rule that a person with cousin-it-specifying-characteristics turns abruptly into a pheasant.
Now we are considering whether two algorithms A1 and A2 that generate distinct streams of numbers “communicate” (whether they’re independent). They are independent if they are generated by different axioms. We would have that there are two sets of axioms, one which possibly generates A1 but not A2, and one which possibly generates A2 but not A1. How do we know that we could find a set of axioms that results in the possibility of only A1 or A2, but not both? I think this is very unlikely, because the possibility of an algorithm already requires a lot of structure, and I doubt you could consistently add to it a set of axioms that specify that A1 is possible but not A2. In our own universe, all the computable algorithms are possibly generated, and this has the symmetry and non-arbitrariness I’ve come to expect from the structure of an entire set of facts deduced from a set of axioms. Sets of axioms don’t result in a fact like ’12345 can move to a position between 1000000 and 1000001 but no other numbers can ever be moved to any other positions’.
It’s conceivable, in contrast, that you have a fact, “numbers can be listed in different orders”. So that moving 12345 would be a possibility but not universally true.
it wouldn’t ever be a universal rule that a person with cousin-it-specifying-characteristics turns abruptly into a pheasant
We agree on that. But why does it have to be a universal rule? In other words, where am I? In the single universe that is 100% lawful, or in one of the myriad chaotic sub-universes embedded within larger lawful structures? For example, the perfectly lawful “universe of all algorithms” contains a lot of entities indistinguishable from me that will horribly disappear the next instant. I’m not insisting on a pheasant—a banana will do as well. If you really believe that all axiomatic structures exist, each passing second of lawfulness should surprise you tremendously.
Sets of axioms don’t result in a fact like ’12345 can move to a position between 1000000 and 1000001 but no other numbers can ever be moved to any other positions’.
Why not? “Axioms” aren’t syntactically distinct from “facts”. You can take any fact and bless it as an axiom.
I can remove contradictions by making up a new rule that specifies which rules take precedence over other ones.
Let’s build a simpler intuition pump: instead of universes, we’ll talk about infinite sequences of integers that can be specified by finite sets of rules. For example, (1, 1, 1, …) is such a sequence. (1, 2, 3, 4, 5, …) is another. Those look regular all right. But this sequence: (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) can also be specified by a finite and self-consistent set of rules, even though something seems to have “swooped down” and changed it in one place. There’s no hard difference between “regular-looking” and “irregular-looking” sequences. All finite sets of rules have equal footing.
Does this make sense to you? Now imagine those integers encoding the time evolution of your toy universe...
It seems to me we’re conflating ‘possible as an output’ and ‘true’ but since I don’t really know what ‘true’ means in this context, let’s conflate them.
The fact that you’ve written the sequence (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) is evidence that it’s the possible output of an algorithm. (Indeed, it was the output of an algorithm you ran.) However, this means that the output was possible (and true) for a subset of the universe. How do you know this rule could be universally true?
I say that it could not be universally true, because it has this property of arbitrariness. I think to answer, ‘what could possibly be universally true?’, you would have to answer the question, ‘what can be deduced as true from nothing?’ or at best, ‘what can be deduced as possibly true from nothing?’ From nothing, the universe might deduce the natural numbers. By definition of what the natural numbers are they have an ordering 1, 2, 3, …, n, n+1, … This ordering really could not be different.
Suppose that the universe had a way of “knowing about” a single element ’12345′ that it places in a new position between 1000000 and 1000001. Simultaneously, it could have placed this element in any position, so universally, you would get a much, much larger structure in which (1, 2, 3, 4, …, 999999, 1000000, 12345, 1000001, 1000002, …) was only a tiny subset.
My point, whether I can figure out how to make it or not, is that the universe doesn’t ‘know about’ 12345, it only knows about all the numbers and evolves this structure universally. You can look at particular components of the structure and observe that individual components seem arbitrary, but the entire structure cannot be.
Imagine a “universe” that consists of all streams of natural numbers that can be specified by algorithms. Is that fundamental and non-arbitrary enough for you? This universe contains many “sub-universes” that cannot communicate, so we can call them “universes” in their own right. One of them is my 12345 sequence, and many others have me spontaneously turn into a pheasant a week from now.
Exactly, yes.
How do you know they don’t communicate? This would be a very non-trivial claim. I’m saying that the set of things that could be independently true (and thus universally true) might be extremely small, and certainly much smaller than the set of possibly-possible things you can think of. Most things we can think of as possible are going to be entangled in ways we aren’t aware of with other truths.
Instead of being where you are thinking of things that could be (‘I turn into a pheasant in 5 minutes’), you would need to turn it upside down and think if there was nothing, what would be true? Perhaps not so many things … perhaps this experienced universe is the only one that was possible. How do we know without developing a theory about what truths self-generate from a void?
Same way I know natural numbers don’t communicate. The output of one algorithm can’t “communicate” with the output of another algorithm, whatever that means.
Each possible universe corresponds to a different set of axioms, right? (If two universes have exactly the same axioms, then they’ll be the same, and the addition of any new axiom that is consistent with but not deducible from the others will make a new universe.)
I’ve maintained all along that arbitrary and weird sets of rules can occur in subsets of the universe, but should not be universally true (true throughout a universe) because there cannot exist a set of axioms that would result in these rules. For example, you can build possibly a machine that turns someone into a pheasant and an extra bag of sand, even in this universe, but it wouldn’t ever be a universal rule that a person with cousin-it-specifying-characteristics turns abruptly into a pheasant.
Now we are considering whether two algorithms A1 and A2 that generate distinct streams of numbers “communicate” (whether they’re independent). They are independent if they are generated by different axioms. We would have that there are two sets of axioms, one which possibly generates A1 but not A2, and one which possibly generates A2 but not A1. How do we know that we could find a set of axioms that results in the possibility of only A1 or A2, but not both? I think this is very unlikely, because the possibility of an algorithm already requires a lot of structure, and I doubt you could consistently add to it a set of axioms that specify that A1 is possible but not A2. In our own universe, all the computable algorithms are possibly generated, and this has the symmetry and non-arbitrariness I’ve come to expect from the structure of an entire set of facts deduced from a set of axioms. Sets of axioms don’t result in a fact like ’12345 can move to a position between 1000000 and 1000001 but no other numbers can ever be moved to any other positions’.
It’s conceivable, in contrast, that you have a fact, “numbers can be listed in different orders”. So that moving 12345 would be a possibility but not universally true.
We agree on that. But why does it have to be a universal rule? In other words, where am I? In the single universe that is 100% lawful, or in one of the myriad chaotic sub-universes embedded within larger lawful structures? For example, the perfectly lawful “universe of all algorithms” contains a lot of entities indistinguishable from me that will horribly disappear the next instant. I’m not insisting on a pheasant—a banana will do as well. If you really believe that all axiomatic structures exist, each passing second of lawfulness should surprise you tremendously.
Why not? “Axioms” aren’t syntactically distinct from “facts”. You can take any fact and bless it as an axiom.
It could mean something that allows them to “communicate”...