Yes, but I don’t think this uses the word ‘exist’ in the same way.
Granting this distinction for the moment, I still think that this points to how a platonist would answer your original question:
[T]o assert the existence of the number 2 seems to me rather vague. What would the world look like if it weren’t so?
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.
In other words, the platonist might argue that formal systems such as PA are accurate (if incomplete) maps of the “territory”, which is the objectively real platonic realm within which abstract mathematical objects exist. If the statement that the number 2 exists were false of the territory, then, since the map is accurate, the corresponding formal statement would not hold within the map. Hence, an empirical implication of the number 2′s not existing would be that you would have the subjective experience of observing the failure of attempts to prove the formal statement that the number 2 exists.
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.
This seems to me rather confused, because it is easy to construct a formal system with which we have precisely that experience. Consider this variant of the Peano Axioms:
0 is a number.
0 has a successor, which is also a number.
The successor of the successor of 0 is not a number. (Alternatively, “is 0”. I think this gives us arithmetic modulo 2.)
(Add the axoms of reflexivity, transitivity, and so on.)
Now clearly, in this formal system I cannot prove the existence of the number two, because its nonexistence is an axiom. Shall I conclude, then, that the number two doesn’t exist, on this account? By what standard are we to judge between formal systems in which 2 is provable, those in which it is disprovable, and those in which it cannot be proved either way? Do we take a vote? Is it a question of appeal to human intuition?
it is easy to construct a formal system with which we have precisely that experience.
I included the qualification “the formal systems that you’re thinking of”. Were you thinking of that formal system when you wrote
There’s a particular set of formal steps I can go through to convince myself that two numbers are twin primes, and a different set of steps which will convince me that some particular pair is the largest such pair, or that there isn’t a largest pair.
?
By what standard are we to judge between formal systems in which 2 is provable, those in which it is disprovable, and those in which it cannot be proved either way? Do we take a vote? Is it a question of appeal to human intuition?
This is a separate question from your original one. To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean [ETA: rather, would imply] that there would be no proof of 2′s existence in that particular system.
But the platonist would also find your new question interesting. For example, when you wrote “There’s a particular set of formal steps …”, how did you come to settle on that particular set of formal steps?
One position would maintain that some particular formal system is implicit in the statement of the twin prime conjecture (TPC). That is, when someone asks “Are there infinitely many twin primes?”, they are speaking in an abbreviated fashion, and they really mean “Does PA prove that there infinitely many twin primes?”, or “Does ZFC prove that …”, or something like that. This position would claim that number-talk has meaning only when the speaker has some specific formal system in mind.
The difficulty with this position is that people seemed to be making meaningful assertions about numbers for thousands of years before settling on any formal systems of arithmetic — indeed before they even had the concept of a formal system. (Euclid’s is presumably too unrigorous to count.) Ancient number-theory texts such as the Introduction to Arithmetic by Nicomachus often didn’t even include anything like what we would call a proof, not even in the sense in which Euclid’s arguments are called proofs. Nicomachus pretty much just flatly asserts number-theoretic propositions, perhaps bolstering his claims with a few examples. Yet somehow readers would reflect on these propositions and agree with them, even though Nicomachus didn’t communicate which formal system he was working within.
Another position would maintain that people absorb some formal system implicitly from their culture, and their number-talk is automatically in terms of that formal system. Even if that culture lacks the concept of a formal system, nonetheless all its number-talk is governed by some formal system.
But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given? Granted, this is necessarily a counterfactual, but . . .
I am not a platonist, but I still can’t call myself a formalist, because I can’t bring myself to declare confidently that the TPC would become meaningless if it were proved undecidable in ZFC. Indeed, even if the TPC happens to be undecidable in every formal system whose axioms would seem to us to be “intuitively correct” of numbers, it still seems to me that our concept of number may suffice to determine a truth value for the above counterfactual. Either that computer would churn out pairs forever, or it wouldn’t.
I included the qualification “the formal systems that you’re thinking of”. Were you thinking of that formal system when you wrote (quote)
I wasn’t, but I rather strongly opine that the word ‘exists’ should not be applied to a state that can change with the vagaries of what formal system I happen to be thinking of. At an absolute minimum, it should be qualified along the lines of “X exists within formal system Y” or “The existence of X is a theorem of formal system Y”. At which point I can return to my original question: What is the formal system of which “God exists” is a theorem or axiom? I also note that “Given axioms X, God exists” is somehow a rather less impressive claims than a floating “God exists”. Yet it’s so much more specific and satisfactory.
To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean that there would be no proof of 2′s existence in that particular system.
This is at least an answer to the question, “What is meant by ‘exist’?”; it gives us a definite procedure for deciding what does and doesn’t exist. But I opine that it’s not a very satisfactory one. Why that formal system and not some other one?
But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given?
I don’t think so, but I’m not sure I understand the relevance, but what I’m objecting to is the word ‘exists’. Suppose you established that the computer was going to print these two numbers and then stop. That is an experimental prediction which we can test. (Updating our belief in the proposition upwards with every second that the computer prints nothing more.) I still don’t see the value in asserting on these grounds that something exists. Why not stick with what is observable, namely that the computer halts, or doesn’t halt?
If you want to say that ‘exists’ is a short form of “the computer halts”, fine; but it does not seem to me that this is what platonists usually intend to say. And, to return to the original problem, it is still completely unclear what “God exists” is shorthand for.
I rather strongly opine that the word ‘exists’ should not be applied to a state that can change with the vagaries of what formal system I happen to be thinking of.
The platonist certainly agrees. The test I described would only work for “accurate” maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean that there would be no proof of 2′s existence in that particular system.
This is at least an answer to the question, “What is meant by ‘exist’?”; it gives us a definite procedure for deciding what does and doesn’t exist. But I opine that it’s not a very satisfactory one. Why that formal system and not some other one?
Actually, I was a little sloppy, there. When speaking on behalf of the platonist, I shouldn’t have written “would mean”, but rather “would imply”. The point is that I wasn’t defining what “the number 2 exists” means, but rather describing what the world would be like if the number 2 didn’t exist.
At any rate, I don’t mean to be giving a “definite procedure”. For the platonist, PA is an accurate, but incomplete, map. Consider that there probably exist physical things of which we will never find any empirical trace. Similarly, the platonist expects that there exist mathematical objects that remain entirely unmapped by our formal systems or intuitions.
I still don’t see the value in asserting on these grounds that something exists.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, “A largest pair of twin primes exists, and this most-recently printed pair is it.”?
Why not stick with what is observable, namely that the computer halts, or doesn’t halt?
(Just to clarify: The program I described never halts, even if the TPC is true. The program continues to run; it continues to brute-force factor progressively larger odd integers. It just never prints anything further to its output tape beyond a certain point.)
The platonist certainly agrees. The test I described would only work for “accurate” maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
Ok, but what is the test which distinguishes between accurate and inaccurate maps? It seems to me that the reasoning here has become circular: It is asserted that the number 2 exists because otherwise, formal system X would be unable to prove it; and also that formal system X is a good test because the number 2 exists. I feel that at this point, you ought to abandon the formal systems and just go for straightforwardly asserting that the number 2 exists in a mystical, intuitive sense which is not open to rational disproof, but which you can use to test formal systems for accuracy.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, “A largest pair of twin primes exists, and this most-recently printed pair is it.”?
Because this does not seem to me to match the meaning of the word ‘exists’. If I say that Planet X exists, I can point to it with a telescope and I expect in principle to be able to travel there. If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it. What am I asserting when I say that a number exists? If it’s that a particular computer will print so many numbers and then stop, then I think this is not the same class of assertion as in the two previous examples, and it ought to have a different word.
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.
Yes and the number 2 exists within the natural numbers, but not within the model your system describes.
I would argue that the same argument MinibearRex uses here to justify his belief on a reality underlying our physical observations. Specifically, if there is no real system underlying our models how do you account for their seeming consistency?
Formal system A: The number 2 exists.
Formal system B: The number 2 does not exist.
I cannot fathom how you can call these systems consistent. Each has a theorem whose negation is a theorem in the other. What possible meaning of ‘consistency’ describes this situation?
Formal system A: The number 2 exists. Formal system B: The number 2 does not exist.
Map of America: Washington, D.C. exists. Map of Europe: Washington, D.C., doesn’t exist.
I cannot fathom how you can call these systems consistent.
Each is a consistent map of its part of the territory. I never said they describe the same part.
As far consistency, would you say PA is as likely to be inconsistent as consistent, because if you believe that PA is just a game of symbols that doesn’t describe anything there seems to be no reason for it to be consistent.
Consistent with what? I believe it is consistent with itself, yes; but then again so is my toy variant with the mod-two arithmetic. If they’re describing a single reality they should be consistent with each other.
Map of America: Washington, D.C. exists. Map of Europe: Washington, D.C., doesn’t exist.
Your analogy fails, because PA and the toy system both agree in describing 0 and 1 as next to each other. But PA asserts that 2 is next to 1, while the toy system explicitly denies that it is so. The map of Europe doesn’t in fact make a claim about the existence of Washington; it just says that if it exists, it’s outside the map. But the toy system makes an explicit claim about the number 2. It’s not that it’s outside the range of the system; the system aggressively asserts that it covers the place where 2 would be if it existed, and also that there ain’t no number there.
Your analogy fails, because PA and the toy system both agree in describing 0 and 1 as next to each other. But PA asserts that 2 is next to 1, while the toy system explicitly denies that it is so. The map of Europe doesn’t in fact make a claim about the existence of Washington; it just says that if it exists, it’s outside the map. But the toy system makes an explicit claim about the number 2. It’s not that it’s outside the range of the system; the system aggressively asserts that it covers the place where 2 would be if it existed, and also that there ain’t no number there.
Granting this distinction for the moment, I still think that this points to how a platonist would answer your original question:
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.
In other words, the platonist might argue that formal systems such as PA are accurate (if incomplete) maps of the “territory”, which is the objectively real platonic realm within which abstract mathematical objects exist. If the statement that the number 2 exists were false of the territory, then, since the map is accurate, the corresponding formal statement would not hold within the map. Hence, an empirical implication of the number 2′s not existing would be that you would have the subjective experience of observing the failure of attempts to prove the formal statement that the number 2 exists.
This seems to me rather confused, because it is easy to construct a formal system with which we have precisely that experience. Consider this variant of the Peano Axioms:
0 is a number.
0 has a successor, which is also a number.
The successor of the successor of 0 is not a number. (Alternatively, “is 0”. I think this gives us arithmetic modulo 2.)
(Add the axoms of reflexivity, transitivity, and so on.)
Now clearly, in this formal system I cannot prove the existence of the number two, because its nonexistence is an axiom. Shall I conclude, then, that the number two doesn’t exist, on this account? By what standard are we to judge between formal systems in which 2 is provable, those in which it is disprovable, and those in which it cannot be proved either way? Do we take a vote? Is it a question of appeal to human intuition?
I included the qualification “the formal systems that you’re thinking of”. Were you thinking of that formal system when you wrote
?
This is a separate question from your original one. To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean [ETA: rather, would imply] that there would be no proof of 2′s existence in that particular system.
But the platonist would also find your new question interesting. For example, when you wrote “There’s a particular set of formal steps …”, how did you come to settle on that particular set of formal steps?
One position would maintain that some particular formal system is implicit in the statement of the twin prime conjecture (TPC). That is, when someone asks “Are there infinitely many twin primes?”, they are speaking in an abbreviated fashion, and they really mean “Does PA prove that there infinitely many twin primes?”, or “Does ZFC prove that …”, or something like that. This position would claim that number-talk has meaning only when the speaker has some specific formal system in mind.
The difficulty with this position is that people seemed to be making meaningful assertions about numbers for thousands of years before settling on any formal systems of arithmetic — indeed before they even had the concept of a formal system. (Euclid’s is presumably too unrigorous to count.) Ancient number-theory texts such as the Introduction to Arithmetic by Nicomachus often didn’t even include anything like what we would call a proof, not even in the sense in which Euclid’s arguments are called proofs. Nicomachus pretty much just flatly asserts number-theoretic propositions, perhaps bolstering his claims with a few examples. Yet somehow readers would reflect on these propositions and agree with them, even though Nicomachus didn’t communicate which formal system he was working within.
Another position would maintain that people absorb some formal system implicitly from their culture, and their number-talk is automatically in terms of that formal system. Even if that culture lacks the concept of a formal system, nonetheless all its number-talk is governed by some formal system.
But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given? Granted, this is necessarily a counterfactual, but . . .
I am not a platonist, but I still can’t call myself a formalist, because I can’t bring myself to declare confidently that the TPC would become meaningless if it were proved undecidable in ZFC. Indeed, even if the TPC happens to be undecidable in every formal system whose axioms would seem to us to be “intuitively correct” of numbers, it still seems to me that our concept of number may suffice to determine a truth value for the above counterfactual. Either that computer would churn out pairs forever, or it wouldn’t.
I wasn’t, but I rather strongly opine that the word ‘exists’ should not be applied to a state that can change with the vagaries of what formal system I happen to be thinking of. At an absolute minimum, it should be qualified along the lines of “X exists within formal system Y” or “The existence of X is a theorem of formal system Y”. At which point I can return to my original question: What is the formal system of which “God exists” is a theorem or axiom? I also note that “Given axioms X, God exists” is somehow a rather less impressive claims than a floating “God exists”. Yet it’s so much more specific and satisfactory.
This is at least an answer to the question, “What is meant by ‘exist’?”; it gives us a definite procedure for deciding what does and doesn’t exist. But I opine that it’s not a very satisfactory one. Why that formal system and not some other one?
I don’t think so, but I’m not sure I understand the relevance, but what I’m objecting to is the word ‘exists’. Suppose you established that the computer was going to print these two numbers and then stop. That is an experimental prediction which we can test. (Updating our belief in the proposition upwards with every second that the computer prints nothing more.) I still don’t see the value in asserting on these grounds that something exists. Why not stick with what is observable, namely that the computer halts, or doesn’t halt?
If you want to say that ‘exists’ is a short form of “the computer halts”, fine; but it does not seem to me that this is what platonists usually intend to say. And, to return to the original problem, it is still completely unclear what “God exists” is shorthand for.
The platonist certainly agrees. The test I described would only work for “accurate” maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
Actually, I was a little sloppy, there. When speaking on behalf of the platonist, I shouldn’t have written “would mean”, but rather “would imply”. The point is that I wasn’t defining what “the number 2 exists” means, but rather describing what the world would be like if the number 2 didn’t exist.
At any rate, I don’t mean to be giving a “definite procedure”. For the platonist, PA is an accurate, but incomplete, map. Consider that there probably exist physical things of which we will never find any empirical trace. Similarly, the platonist expects that there exist mathematical objects that remain entirely unmapped by our formal systems or intuitions.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, “A largest pair of twin primes exists, and this most-recently printed pair is it.”?
(Just to clarify: The program I described never halts, even if the TPC is true. The program continues to run; it continues to brute-force factor progressively larger odd integers. It just never prints anything further to its output tape beyond a certain point.)
Ok, but what is the test which distinguishes between accurate and inaccurate maps? It seems to me that the reasoning here has become circular: It is asserted that the number 2 exists because otherwise, formal system X would be unable to prove it; and also that formal system X is a good test because the number 2 exists. I feel that at this point, you ought to abandon the formal systems and just go for straightforwardly asserting that the number 2 exists in a mystical, intuitive sense which is not open to rational disproof, but which you can use to test formal systems for accuracy.
Because this does not seem to me to match the meaning of the word ‘exists’. If I say that Planet X exists, I can point to it with a telescope and I expect in principle to be able to travel there. If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it. What am I asserting when I say that a number exists? If it’s that a particular computer will print so many numbers and then stop, then I think this is not the same class of assertion as in the two previous examples, and it ought to have a different word.
Both maps are accurate maps of different parts of the territory.
Yet they both make assertions about the number 0, its successor, and its successor’s successor.
They use the word ‘successor’ to mean slightly different things.
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.
This is a universal counterargument against saying that anything besides what we are currently observing exists.
Yes and the number 2 exists within the natural numbers, but not within the model your system describes.
I would argue that the same argument MinibearRex uses here to justify his belief on a reality underlying our physical observations. Specifically, if there is no real system underlying our models how do you account for their seeming consistency?
What consistency? You just acknowledged that there are formal systems in which 2 doesn’t exist.
The fact that the formal systems are consistent.
And there are planets on which humans don’t exist. I don’t see how this is inconsistent.
Formal system A: The number 2 exists. Formal system B: The number 2 does not exist.
I cannot fathom how you can call these systems consistent. Each has a theorem whose negation is a theorem in the other. What possible meaning of ‘consistency’ describes this situation?
Map of America: Washington, D.C. exists. Map of Europe: Washington, D.C., doesn’t exist.
Each is a consistent map of its part of the territory. I never said they describe the same part.
As far consistency, would you say PA is as likely to be inconsistent as consistent, because if you believe that PA is just a game of symbols that doesn’t describe anything there seems to be no reason for it to be consistent.
Consistent with what? I believe it is consistent with itself, yes; but then again so is my toy variant with the mod-two arithmetic. If they’re describing a single reality they should be consistent with each other.
Your analogy fails, because PA and the toy system both agree in describing 0 and 1 as next to each other. But PA asserts that 2 is next to 1, while the toy system explicitly denies that it is so. The map of Europe doesn’t in fact make a claim about the existence of Washington; it just says that if it exists, it’s outside the map. But the toy system makes an explicit claim about the number 2. It’s not that it’s outside the range of the system; the system aggressively asserts that it covers the place where 2 would be if it existed, and also that there ain’t no number there.
Can you tell me your basis for this belief?
Answered here.