I didn’t see myself as defending platonism, so much as defending a certain kind of existence-talk that I, along with the platonists, think is legitimate.
Ok, but here is your original reply:
The platonist might reply: If the number 2 didn’t exist in the platonic sense, then one implication would be that you would be unable to construct proofs of that number’s existence within the formal systems that you’re thinking of.
And all your replies since have taken the form “the platonist might say”. Can I suggest that you should defend the theory you actually believe in? At this point you seem to agree with me that theorems within some particular formal theory is not a good reason to say that numbers exist. What then do you mean by asserting the existence of the number 2? No more platonism, if you please, since you don’t believe in it.
Couldn’t an empirically well-supported theory give high probability to the claim that there exists a species X whose future and past lightcones never intersect yours, and which is not accessible to you by any closed timelike curves or other means of extending the lightcone?
No, actually, I don’t see how it could. If, by construction, I never, either in past or future, interact with any of the species or with anything that it has interacted with, then I don’t see how I can get an empirically supported theory of their existence.
And besides, wouldn’t it be kind of weird if your ability to use the word “exists” were constrained by how sophisticated your physics is?
And all your replies since have taken the form “the platonist might say”. Can I suggest that you should defend the theory you actually believe in?
I haven’t been able to come up with a full-fledged theory of what mathematics is about that I’m happy with. I can say some vague things, but there is no reason to burden you with their vagueness. I spoke from the perspective of platonism so that at least there would be a concrete theory of mathematical meaning on the table. I have no alternative concrete theory to offer in its place.
Nonetheless, all the difficulties I raised above about trying to avoid talk about the existence of the number 2 are difficulties that I really think are problems for your position. That is, they are issues that keep me from being convinced of your view.
I don’t see a principled way to avoid talking about the existence of a largest pair of twin primes in a sense that is independent of any particular formal system. You’ve said that you would allow talk about such a pair’s existence only if such talk amounted to statements about what certain sequences of computations would yield. However, this appears to commit you to the existence of sequences of computations. It doesn’t seem helpful to reduce this sense of existence to derivations within formal systems, because that commits you to the existence of formal systems — an existence, moreover, that appears to be in a sense that is independent of any particular formal system, lest an infinite regression drain all appearance of meaning from any kind of existence-talk.
Compared to all these abstruse abstract objects (sequences, computations, and formal systems), talk of the existence of numbers seems very innocent to me.
Furthermore, as I’m arguing in the “physics” thread of our conversation, existence-talk about physical objects seems to me to suffer from some of the same obscurities that existence talk about mathematical objects does: Namely, in neither case does a purely positivistic reduction of “exists” really work.
I admit that I’m confused about what “exists”, at bottom, really means. But I don’t know how to get by without speaking of the existence of numbers, any more than I know how to get by without talking about the existence of physical things.
No, actually, I don’t see how it could. If, by construction, I never, either in past or future, interact with any of the species or with anything that it has interacted with, then I don’t see how I can get an empirically supported theory of their existence.
Could we never have empirical support for a Tegmark Level I multiverse? More to the point, isn’t it at least meaningful to pose the possibility of such a multiverse, even though it amounts to suggesting the existence of many things with which you never can have and never could have had any causal interaction?
Ok, just use the past interaction then.
Past interaction were ruled out in my scenario (“while you cannot because you (let’s say) believe in a universe in which it is and was always impossible for you to interact causally with anything that far away.”).
I spoke sloppily. I meant that I would use ‘exist’ about a species I had interacted with in the past, not one I could in-principle interact with by breaking known laws of physics.
Could we never have empirical support for a Tegmark Level I multiverse? More to the point, isn’t it at least meaningful to pose the possibility of such a multiverse, even though it amounts to suggesting the existence of many things with which you never can have and never could have had any causal interaction?
This gives me a new idea, actually. If you assert the existence of such a multiverse, you are saying that things like us exist. They have consciousness, they interact with objects, in short they have all the hallmarks of the existence of physical things. When I say that such a thing exists, I’m using the word in the same sense as when I speak of a rock. With what does a number interact? Nothing. If you allow interventions from outside the Matrix, I could interact with humans that are causally separated from my past and future. But even that power will not allow me to interact with the number 2; I cannot affect it, or it me, in any sense.
With what does a number interact? Nothing. If you allow interventions from outside the Matrix, I could interact with humans that are causally separated from my past and future. But even that power will not allow me to interact with the number 2; I cannot affect it, or it me, in any sense.
The post you link to talks about controlling a computer program by making decisions, in particular the decision to one-box. I think that this would sound rather less impressive if it were converted to talking about controlling an FPS by moving the mouse and pressing the space bar, which is functionally equivalent.
Those devices take in inputs. The post is talking about controlling programs that take no inputs. In other words, they are fixed mathematical structures.
Just because the input is a fixed simulation of the agent’s decision doesn’t mean the calculation as a whole has no inputs! In particular, it has whatever inputs are decisive for the agent itself. An agent that gets whacked with a stick every time it one-boxes is quite likely to make a different decision from one with the same algorithm but working on data that doesn’t include stick-whackings. It’s not sufficient to specify the laws of physics, you have to know the boundary conditions as well.
Ok, but here is your original reply:
And all your replies since have taken the form “the platonist might say”. Can I suggest that you should defend the theory you actually believe in? At this point you seem to agree with me that theorems within some particular formal theory is not a good reason to say that numbers exist. What then do you mean by asserting the existence of the number 2? No more platonism, if you please, since you don’t believe in it.
No, actually, I don’t see how it could. If, by construction, I never, either in past or future, interact with any of the species or with anything that it has interacted with, then I don’t see how I can get an empirically supported theory of their existence.
Ok, just use the past interaction then.
I haven’t been able to come up with a full-fledged theory of what mathematics is about that I’m happy with. I can say some vague things, but there is no reason to burden you with their vagueness. I spoke from the perspective of platonism so that at least there would be a concrete theory of mathematical meaning on the table. I have no alternative concrete theory to offer in its place.
Nonetheless, all the difficulties I raised above about trying to avoid talk about the existence of the number 2 are difficulties that I really think are problems for your position. That is, they are issues that keep me from being convinced of your view.
I don’t see a principled way to avoid talking about the existence of a largest pair of twin primes in a sense that is independent of any particular formal system. You’ve said that you would allow talk about such a pair’s existence only if such talk amounted to statements about what certain sequences of computations would yield. However, this appears to commit you to the existence of sequences of computations. It doesn’t seem helpful to reduce this sense of existence to derivations within formal systems, because that commits you to the existence of formal systems — an existence, moreover, that appears to be in a sense that is independent of any particular formal system, lest an infinite regression drain all appearance of meaning from any kind of existence-talk.
Compared to all these abstruse abstract objects (sequences, computations, and formal systems), talk of the existence of numbers seems very innocent to me.
Furthermore, as I’m arguing in the “physics” thread of our conversation, existence-talk about physical objects seems to me to suffer from some of the same obscurities that existence talk about mathematical objects does: Namely, in neither case does a purely positivistic reduction of “exists” really work.
I admit that I’m confused about what “exists”, at bottom, really means. But I don’t know how to get by without speaking of the existence of numbers, any more than I know how to get by without talking about the existence of physical things.
Could we never have empirical support for a Tegmark Level I multiverse? More to the point, isn’t it at least meaningful to pose the possibility of such a multiverse, even though it amounts to suggesting the existence of many things with which you never can have and never could have had any causal interaction?
Past interaction were ruled out in my scenario (“while you cannot because you (let’s say) believe in a universe in which it is and was always impossible for you to interact causally with anything that far away.”).
I spoke sloppily. I meant that I would use ‘exist’ about a species I had interacted with in the past, not one I could in-principle interact with by breaking known laws of physics.
This gives me a new idea, actually. If you assert the existence of such a multiverse, you are saying that things like us exist. They have consciousness, they interact with objects, in short they have all the hallmarks of the existence of physical things. When I say that such a thing exists, I’m using the word in the same sense as when I speak of a rock. With what does a number interact? Nothing. If you allow interventions from outside the Matrix, I could interact with humans that are causally separated from my past and future. But even that power will not allow me to interact with the number 2; I cannot affect it, or it me, in any sense.
Taboo ‘interact’.
The decision-theory people on LW talk about agents controlling abstract mathematical entities in a way that I cannot so easily dismiss.
The post you link to talks about controlling a computer program by making decisions, in particular the decision to one-box. I think that this would sound rather less impressive if it were converted to talking about controlling an FPS by moving the mouse and pressing the space bar, which is functionally equivalent.
Those devices take in inputs. The post is talking about controlling programs that take no inputs. In other words, they are fixed mathematical structures.
Just because the input is a fixed simulation of the agent’s decision doesn’t mean the calculation as a whole has no inputs! In particular, it has whatever inputs are decisive for the agent itself. An agent that gets whacked with a stick every time it one-boxes is quite likely to make a different decision from one with the same algorithm but working on data that doesn’t include stick-whackings. It’s not sufficient to specify the laws of physics, you have to know the boundary conditions as well.