Yes, I understand that just one of P and P2 would be taken as representative. And I’m saying that then the question of whether X (the thing computed by P and P2) “exists” may have a different answer depending on whether we arbitrarily picked P or P2, and that that seems unsatisfactory.
It would not have a different answer. If P is chosen for Y, and X and Y belong to the same equivalence class, then X would be computed as P, and vice versa for P2.
I think one of us is misunderstanding the other. I’m not saying that two equivalent things would be treated differently. That is, I’m not concerned that we would get different answers to “does X exist?” and “does Y exist?” with X and Y being equivalent.
My concern is a different one. In order to decide whether (say) X exists, we have to make an arbitrary choice of whether to use P or P2. We will then use the same choice for Y as well, so X and Y will come out the same way, and that’s fine. But that arbitrary choice will make a difference to whether or not we say that X and Y exist, and that is what bothers me.
(I said much the same thing six comments upthread and it seems like I was misunderstood then too. Or perhaps I have been misunderstanding everything you’ve been saying since then :-).)
But that arbitrary choice will make a difference to whether or not we say that X and Y exist, and that is what bothers me.
I do not understand how this follows from:
In order to decide whether (say) X exists, we have to make an arbitrary choice of whether to use P or P2. We will then use the same choice for Y as well, so X and Y will come out the same way, and that’s fine.
If there’s a problem, then it is the inference you’re making from the second quote to the first. That is the cause of the misunderstanding. I would appreciate it if you explained why you think the arbitrary choice makes a difference whether an object exists (that is not supposed to happen, and I designed it so it wouldn’t).
I am not sure what to say that might clarify further, but let me try again.
First of all, I am not making an inference from the second quote to the first. The second one is not there to support the first one. It is there because you seemed to be misunderstanding me, and I wanted to deal with the misunderstanding.
So let me see what I can say to support my objection.
First of all, your definition says (roughly) that the criterion for X to “exist” is: let P be “the” minimal program that implements X; then X exists iff P appears somewhere in “the output of G(Q)”. Here G(Q) is a computer program that implements our universe given input Q; in other words, its output is (some sort of symbolic representation of) our universe.
Question: does the above paragraph correctly describe your intention?
Now, suppose there are in fact two minimal programs that implement X. Call them P and P2. Your definition says that we are supposed to pick one of them. (We are supposed to do this consistently, in the sense that if we pick P rather than P2, we should also pick P rather than P2 for anything else implemented by these programs. This is not directly relevant here, though.)
Question: does the above paragraph correctly describe your intention?
If the answers to my two italicized questions are correct, then:
If we choose P, then X “exists” iff P appears somewhere in (a symbolic representation of) our universe. If we choose P2, then X “exists” iff P2 appears somewhere in (a symbolic representation of) our universe.
There is no reason why these two propositions should be equivalent. So whether we choose P or P2 may make a difference to whether or not X “exists”.
(This is, just to reiterate something I hope I made clear above, not my principal objection to your proposal as I understand it. One could work around this objection by saying e.g. that X “exists” iff there is some minimal program implementing X that appears in G(Q). My more fundamental objection is that it seems perfectly obvious to me that whether P appears in G(Q) has nothing whatever to do with whether X exists, because there is no reason why the universe should contain programs implementing all its objects.)
If we choose P, then X “exists” iff P appears somewhere in (a symbolic representation of) our universe. If we choose P2, then X “exists” iff P2 appears somewhere in (a symbolic representation of) our universe.
There is no reason why these two propositions should be equivalent. So whether we choose P or P2 may make a difference to whether or not X “exists”.
As I understand this, the two propositions are equivalent. We do not arbitrarily pick P or P2 (we assume that one is picked and is picked consistently). What this means is that the G(TM) will also pick P or P2 consistently. If G(TM) outputs P, then it would never output P2 and vice versa. Only one of the members of the equivalence class become the shortest program, and that member represents the entire class everytime the class is invoked. So the shortest program will be computed by G consistently when simulating our universe.
My more fundamental objection is that it seems perfectly obvious to me that whether P appears in G(Q) has nothing whatever to do with whether X exists, because there is no reason why the universe should contain programs implementing all its objects.)
Yes, I agree that the universe can be simulated (or, more precisely, my guess is that it can be, the available scientific evidence suggests that it probably can be, and it’s a convenient working hypothesis).
I’m afraid I don’t at all understand your argument here. “If G(TM) outputs P, then it would never output P2 and vice versa”—I have no inkling why that should be true. Why do you believe that G(TM) cannot output both of them on different occasions?
Yes, I understand that just one of P and P2 would be taken as representative. And I’m saying that then the question of whether X (the thing computed by P and P2) “exists” may have a different answer depending on whether we arbitrarily picked P or P2, and that that seems unsatisfactory.
It would not have a different answer. If P is chosen for Y, and X and Y belong to the same equivalence class, then X would be computed as P, and vice versa for P2.
P.S: sorry for the late reply.
I think one of us is misunderstanding the other. I’m not saying that two equivalent things would be treated differently. That is, I’m not concerned that we would get different answers to “does X exist?” and “does Y exist?” with X and Y being equivalent.
My concern is a different one. In order to decide whether (say) X exists, we have to make an arbitrary choice of whether to use P or P2. We will then use the same choice for Y as well, so X and Y will come out the same way, and that’s fine. But that arbitrary choice will make a difference to whether or not we say that X and Y exist, and that is what bothers me.
(I said much the same thing six comments upthread and it seems like I was misunderstood then too. Or perhaps I have been misunderstanding everything you’ve been saying since then :-).)
I do not understand how this follows from:
In order to decide whether (say) X exists, we have to make an arbitrary choice of whether to use P or P2. We will then use the same choice for Y as well, so X and Y will come out the same way, and that’s fine.
If there’s a problem, then it is the inference you’re making from the second quote to the first. That is the cause of the misunderstanding. I would appreciate it if you explained why you think the arbitrary choice makes a difference whether an object exists (that is not supposed to happen, and I designed it so it wouldn’t).
I am not sure what to say that might clarify further, but let me try again.
First of all, I am not making an inference from the second quote to the first. The second one is not there to support the first one. It is there because you seemed to be misunderstanding me, and I wanted to deal with the misunderstanding.
So let me see what I can say to support my objection.
First of all, your definition says (roughly) that the criterion for X to “exist” is: let P be “the” minimal program that implements X; then X exists iff P appears somewhere in “the output of G(Q)”. Here G(Q) is a computer program that implements our universe given input Q; in other words, its output is (some sort of symbolic representation of) our universe.
Question: does the above paragraph correctly describe your intention?
Now, suppose there are in fact two minimal programs that implement X. Call them P and P2. Your definition says that we are supposed to pick one of them. (We are supposed to do this consistently, in the sense that if we pick P rather than P2, we should also pick P rather than P2 for anything else implemented by these programs. This is not directly relevant here, though.)
Question: does the above paragraph correctly describe your intention?
If the answers to my two italicized questions are correct, then:
If we choose P, then X “exists” iff P appears somewhere in (a symbolic representation of) our universe. If we choose P2, then X “exists” iff P2 appears somewhere in (a symbolic representation of) our universe.
There is no reason why these two propositions should be equivalent. So whether we choose P or P2 may make a difference to whether or not X “exists”.
(This is, just to reiterate something I hope I made clear above, not my principal objection to your proposal as I understand it. One could work around this objection by saying e.g. that X “exists” iff there is some minimal program implementing X that appears in G(Q). My more fundamental objection is that it seems perfectly obvious to me that whether P appears in G(Q) has nothing whatever to do with whether X exists, because there is no reason why the universe should contain programs implementing all its objects.)
As I understand this, the two propositions are equivalent. We do not arbitrarily pick P or P2 (we assume that one is picked and is picked consistently). What this means is that the G(TM) will also pick P or P2 consistently. If G(TM) outputs P, then it would never output P2 and vice versa. Only one of the members of the equivalence class become the shortest program, and that member represents the entire class everytime the class is invoked. So the shortest program will be computed by G consistently when simulating our universe.
Do you agree that the universe can be simulated?
(Sorry about the long delay in replying.)
Yes, I agree that the universe can be simulated (or, more precisely, my guess is that it can be, the available scientific evidence suggests that it probably can be, and it’s a convenient working hypothesis).
I’m afraid I don’t at all understand your argument here. “If G(TM) outputs P, then it would never output P2 and vice versa”—I have no inkling why that should be true. Why do you believe that G(TM) cannot output both of them on different occasions?