If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it.
I was a little sloppy on this point earlier, so I want to be more careful about it now:
Are you saying that “Species X exists” means exactly the same thing as “I could shoot and eat X in principle”? Or are you just saying that, if you show that you can shoot and eat X, you’ve shown that X exists? For example, is it meaningless to assert the existence of a species outside of your lightcone?
What am I asserting when I say that a number exists? If it’s that a particular computer will print so many numbers …
It becomes very tricky to talk without appearing to commit yourself to the existence of numbers.
For example, you say above that the computer is printing numbers. How is it doing that if numbers don’t exist?
That might seem like nitpicking, and you probably could find a perfectly adequate rewording of that sentence that avoids even the appearance of implying the existence of numbers. But it has proved very difficult to give a fully satisfying nominalist rewording of all talk that is ostensibly about abstract objects.
ETA: The question you raise about how we could know, on a platonic account, that a given formal system is an accurate map of the natural numbers is a good one. It’s a large part of why platonism is a nonstarter for me. But I don’t think that it’s incoherent in the same way that you seem to.
For example, you say above that the computer is printing numbers. How is it doing that if numbers don’t exist?
Well then, let it arrange groups of pebbles instead.
The question you raise about how we could know, on a platonic account, that a given formal system is an accurate map of the natural numbers is a good one. It’s a large part of why platonism is a nonstarter for me. But I don’t think that it’s incoherent in the same way that you seem to.
I gently suggest that you need to check your association maps, here. If you can give no account of this very fundamental thing, how we know that particular formal systems are the right ones to use, then isn’t it time to go from “platonism is a nonstarter for me” to “platonism is wrong”?
Are you saying that “Species X exists” means exactly the same thing as “I could shoot and eat X in principle”? Or are you just saying that, if you show that you can shoot and eat X, you’ve shown that X exists? For example, is it meaningless to assert the existence of a species outside of your lightcone?
How would I learn about such a species? It may not be meaningless, but I don’t see how to connect it to any experience.
I gently suggest that you need to check your association maps, here.
What do you mean by an “associate map” [ETA: oops, “association map”] in this context?
If you can give no account of this very fundamental thing, how we know that particular formal systems are the right ones to use, then isn’t it time to go from “platonism is a nonstarter for me” to “platonism is wrong”?
Unfortunately, there are many very fundamental things for which I am unable to give any account worth the time. Strangely, it seems that the more fundamental something is, the more difficult it is to account for it satisfactorily.
At any rate, you can take “platonism is a nonstarter for me” to imply “I think that platonism is wrong”. (What else would “nonstarter” mean?)
How would I learn about such a species? It may not be meaningless, but I don’t see how to connect it to any experience.
As in Belief in the Implied Invisible, a physical theory with strong empirical support could commit you to believing in such a species with high probability.
What do you mean by an “associate map” in this context?
I mean that you should check that you’re not compartmentalising your beliefs. If we don’t regularly test the implications and associations between our beliefs, we can end up asserting contradictory things, like the theist scientist. That said, this:
At any rate, you can take “platonism is a nonstarter for me” to imply “I think that platonism is wrong”.
seems to indicate that the exercise isn’t necessary for you here.
(What else would “nonstarter” mean?)
At any rate it apparently allows you to defend platonism to some extent. Perhaps ‘non-finisher’ would be a better term?
As in Belief in the Implied Invisible, a physical theory with strong empirical support could commit you to believing in such a species with high probability.
Yes, but then I could broaden my shooting definition slightly and say “At one time it was possible to shoot this species.”; in other words I would give the past empirie that convinced me of the species’ existence. Or alternatively, I could go for “in principle” and say that we just need some closed timelike curvies, or other means of extending the lightcone.
At any rate it apparently allows you to defend platonism to some extent.
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. I agree with them that one should be able to consider the possibility that a largest pair of twin primes exists. Furthermore, it should be possible to do this without having a formal procedure for deciding the question in mind, even implicitly.
Yes, but then I could broaden my shooting definition slightly and say “At one time it was possible to shoot this species.”; in other words I would give the past empirie that convinced me of the species’ existence. Or alternatively, I could go for “in principle” and say that we just need some closed timelike curvies, or other means of extending the lightcone.
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?
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? It would seem to follow that someone who believes in a classical Newtonian universe allowing arbitrarily fast travel can legitimately ponder the possibility that dragons exist 10^100 light years away, 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 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.
I was a little sloppy on this point earlier, so I want to be more careful about it now:
Are you saying that “Species X exists” means exactly the same thing as “I could shoot and eat X in principle”? Or are you just saying that, if you show that you can shoot and eat X, you’ve shown that X exists? For example, is it meaningless to assert the existence of a species outside of your lightcone?
It becomes very tricky to talk without appearing to commit yourself to the existence of numbers.
For example, you say above that the computer is printing numbers. How is it doing that if numbers don’t exist?
That might seem like nitpicking, and you probably could find a perfectly adequate rewording of that sentence that avoids even the appearance of implying the existence of numbers. But it has proved very difficult to give a fully satisfying nominalist rewording of all talk that is ostensibly about abstract objects.
ETA: The question you raise about how we could know, on a platonic account, that a given formal system is an accurate map of the natural numbers is a good one. It’s a large part of why platonism is a nonstarter for me. But I don’t think that it’s incoherent in the same way that you seem to.
Well then, let it arrange groups of pebbles instead.
I gently suggest that you need to check your association maps, here. If you can give no account of this very fundamental thing, how we know that particular formal systems are the right ones to use, then isn’t it time to go from “platonism is a nonstarter for me” to “platonism is wrong”?
How would I learn about such a species? It may not be meaningless, but I don’t see how to connect it to any experience.
What do you mean by an “associate map” [ETA: oops, “association map”] in this context?
Unfortunately, there are many very fundamental things for which I am unable to give any account worth the time. Strangely, it seems that the more fundamental something is, the more difficult it is to account for it satisfactorily.
At any rate, you can take “platonism is a nonstarter for me” to imply “I think that platonism is wrong”. (What else would “nonstarter” mean?)
As in Belief in the Implied Invisible, a physical theory with strong empirical support could commit you to believing in such a species with high probability.
I mean that you should check that you’re not compartmentalising your beliefs. If we don’t regularly test the implications and associations between our beliefs, we can end up asserting contradictory things, like the theist scientist. That said, this:
seems to indicate that the exercise isn’t necessary for you here.
At any rate it apparently allows you to defend platonism to some extent. Perhaps ‘non-finisher’ would be a better term?
Yes, but then I could broaden my shooting definition slightly and say “At one time it was possible to shoot this species.”; in other words I would give the past empirie that convinced me of the species’ existence. Or alternatively, I could go for “in principle” and say that we just need some closed timelike curvies, or other means of extending the lightcone.
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. I agree with them that one should be able to consider the possibility that a largest pair of twin primes exists. Furthermore, it should be possible to do this without having a formal procedure for deciding the question in mind, even implicitly.
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?
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? It would seem to follow that someone who believes in a classical Newtonian universe allowing arbitrarily fast travel can legitimately ponder the possibility that dragons exist 10^100 light years away, 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.
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.