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.
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”...