The key is the qualification “from the bare description, ‘quarks.’”
To elaborate—JoshuaZ’s comment brought this home—you can distinguish infinite sets by their cardinality or by their subset/superset relationship, and these are independent. The reasoning about quarks brackets all knowledge about the distinctions between quarks that could be used to establish a set/superset relationship.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them” which I think is what you’re doing.
If there are infinitely many quarks, then we can form infinite sets of quarks. One of these sets is the set of all quarks. This set is infinite, includes all quarks, and there are no quarks it doesn’t include, and saying anything else is patent nonsense whether you’re talking about quarks, integers, or kittens.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them”
Sets with different elements are different. But, unfortunately for actually realized infinities, you can argue that two sets with different elements are the same when those infinite sets are actually realized—but only because actually realized infinities are incoherent. That you can argue both sides, contradicted only by the other side, is what makes actual infinity incoherent.
You can’t defeat an argument purporting to show a contradiction by simply upholding one side; you can’t deny me the argument that the two sets are the same (as part of that argument to contradiction) simply based on a separate argument that they’re different.
for actually realized infinities, you can argue that two sets with different elements are the same
Suppose I restate your argument for integers instead of quarks:
“If there are infinitely many integers, then I can form an infinite set of integers. That set includes all the integers, since there can be no set of the same cardinality that’s greater and because, from the bare description, “integers,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers. [I don’t follow this sentence, so I’ve just copied it.]. But that set does not include all the integers because the existence of other integers outside the set is consistent with the set’s defining requirement that it contain infinitely many elements.”
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between an abstract infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
Also, regarding this key sentence:
That set includes all the quarks in the universe, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of quarks.
I don’t follow this sentence, I didn’t follow the clarification you made three posts up. Perhaps you could expand this sentence into a paragraph or two that a five year old could understand?
Suppose I restate your argument for integers instead of quarks...
We don’t need to assume there are infinitely many integers, only that integers are unlimited. Some Platonists may think that an infinite set of integers is realized, and I think the arguments does pertain to that claim.
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between [a potential] infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
The distinction is relevant to why I have no quarrel with potential infinities as such.
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
No. It’s only the case if (per stipulation) you know nothing about properties that distinguish one quark from another. Then, the only way you can form an infinite set of quarks is by taking all of them. So, I’m not assuming that any infinite set of quarks I can form is the only infinite set of quarks I can form; I’m setting up the problem so there’s only one way to form an infinite set of quarks. Any set conforming to that description “should” be the only set.
Perhaps you could expand this sentence:
That set includes all the quarks, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers.
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately. This is because there are only two ways that infinite subsets can be distinguished from their supersets: 1) the subset is of lower cardinality than the superset or 2) the elements are distinguishable to create a logical superset/set relationship (such as exists in quarks/upside-down quarks).
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately.
OK, suppose I grant this. I now feel like I might be able to formulate your argument in my own words. Here’s an attempt; let me know if and when it diverges from what you’re actually arguing.
--
“Suppose I have sworn to give up the hateful practice of discriminating between quarks based on their differences. Henceforth I shall treat all quarks as utterly indistinguishable from one another. Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
“I inspect the set S and try to see whether it’s different from the set of all quarks, which we call Q. First I look at the cardinalities of S and Q. If their cardinalities were different, then obviously S and Q would be different sets. But their cardinalities are the same. Next I look for a quark that is contained in Q, but not contained in S. If there were such an element, then obviously S and Q would be different sets. But in order to successfully find such an element, I would have to make use of the distinctions between quarks. After all, how would I know that a given quark was in Q, but not in S? I would have to show that the quark in Q was distinct from each quark in S, but I have agreed to regard all quarks as indistinguishable. Therefore my search for an element of Q that is not in S will fail. I conclude that the set S is the same as the set Q. That is the set you gave me must be the set of all quarks.
“But this conclusion is obviously wrong. All I asked you for was an infinite set of quarks. There are many infinite sets of quarks, not all of which are the same as Q, the set of all quarks. You might have left some quarks out of S, and still provided me with an infinite set of quarks, which was all I asked for.
“Therefore we have a contradiction: I have proved something that is not necessarily true. Therefore the set of quarks cannot be infinite.”
--
The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets.
I take issue with your translation at only a single point:
Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
My version contains a further constraint: When you ask me to bring you an infinite set of quarks, you instruct me to be as blind as you to the features that distinguish between quarks.
The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets. [emphasis added.]
The_Duck tells metaphysicist to gather together an infinite set of quarks while remaining blind to their individuality. Metaphysicist, having no distinctions on which to carve infinite subsets, can respond to this request in only one way; include every quark. (I want to resist calling this the “set of all quarks,” because the incoherence of that concept with infinite quarks is what I argue.) The_Duck then goes out and finds another quark, and scolds metaphysicist, “You missed one.”
The_Duck is unjustified in criticizing metaphysicist, who must have picked “all the quarks,” given that metaphysicist succeeded—without knowing of any proper subsets—in assembling an infinite set . Having “selected all the quarks” doesn’t preclude finding another when they’re infinite in number and the only criterion for success is the number.
You will say that there is a fact of the matter as to whether the first set I assembled was all the quarks. Unblind yourself to the quarks’ individuating features, you say, and you get an underlying reality where the sets are different. I agree, but I think a more limited point suffices. When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark. There’s no way for me to distinguish the two sets. The contradiction is that following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
I take issue with your translation at only a single point:
I’m making progress then. :)
When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark.
No. If what you gathered is a proper subset of what you could have gathered, then you didn’t gather all the quarks, and you’re not justified in claiming that you did. How did you decide to leave out that one other quark? You must have made a distinction between it and the others that you did gather.
There’s no way for me to distinguish the two sets.
Of course there is. The superset contains a quark that the subset doesn’t. If you refuse to notice the differences that single that quark out from the others, that’s your loss.
I think that maybe you’re trying not to distinguish between quarks, but are implicitly distinguishing between “quarks that you know about” and “quarks that you don’t know about.” So you might assemble all the quarks you know about—an infinite number—and not have any evidence that this isn’t all the quarks there are. But later, you worry, you might find some other quarks that you didn’t know about before, so that your original set didn’t actually contain all quarks. This is not contradictory. If there was a chance that there existed quarks you didn’t know about, then you weren’t justified in saying that you had gathered all the quarks.
following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
It does. If you’re not at the top of the hierarchy, you haven’t gathered all the quarks. And you can’t justify claiming that you’re at the top of the hierarchy by blinding yourself to evidence that would prove otherwise.
The key is the qualification “from the bare description, ‘quarks.’”
To elaborate—JoshuaZ’s comment brought this home—you can distinguish infinite sets by their cardinality or by their subset/superset relationship, and these are independent. The reasoning about quarks brackets all knowledge about the distinctions between quarks that could be used to establish a set/superset relationship.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them” which I think is what you’re doing.
If there are infinitely many quarks, then we can form infinite sets of quarks. One of these sets is the set of all quarks. This set is infinite, includes all quarks, and there are no quarks it doesn’t include, and saying anything else is patent nonsense whether you’re talking about quarks, integers, or kittens.
Sets with different elements are different. But, unfortunately for actually realized infinities, you can argue that two sets with different elements are the same when those infinite sets are actually realized—but only because actually realized infinities are incoherent. That you can argue both sides, contradicted only by the other side, is what makes actual infinity incoherent.
You can’t defeat an argument purporting to show a contradiction by simply upholding one side; you can’t deny me the argument that the two sets are the same (as part of that argument to contradiction) simply based on a separate argument that they’re different.
Suppose I restate your argument for integers instead of quarks:
“If there are infinitely many integers, then I can form an infinite set of integers. That set includes all the integers, since there can be no set of the same cardinality that’s greater and because, from the bare description, “integers,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers. [I don’t follow this sentence, so I’ve just copied it.]. But that set does not include all the integers because the existence of other integers outside the set is consistent with the set’s defining requirement that it contain infinitely many elements.”
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between an abstract infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
Also, regarding this key sentence:
I don’t follow this sentence, I didn’t follow the clarification you made three posts up. Perhaps you could expand this sentence into a paragraph or two that a five year old could understand?
We don’t need to assume there are infinitely many integers, only that integers are unlimited. Some Platonists may think that an infinite set of integers is realized, and I think the arguments does pertain to that claim.
The distinction is relevant to why I have no quarrel with potential infinities as such.
No. It’s only the case if (per stipulation) you know nothing about properties that distinguish one quark from another. Then, the only way you can form an infinite set of quarks is by taking all of them. So, I’m not assuming that any infinite set of quarks I can form is the only infinite set of quarks I can form; I’m setting up the problem so there’s only one way to form an infinite set of quarks. Any set conforming to that description “should” be the only set.
That set includes all the quarks, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers.
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately. This is because there are only two ways that infinite subsets can be distinguished from their supersets: 1) the subset is of lower cardinality than the superset or 2) the elements are distinguishable to create a logical superset/set relationship (such as exists in quarks/upside-down quarks).
OK, suppose I grant this. I now feel like I might be able to formulate your argument in my own words. Here’s an attempt; let me know if and when it diverges from what you’re actually arguing.
--
“Suppose I have sworn to give up the hateful practice of discriminating between quarks based on their differences. Henceforth I shall treat all quarks as utterly indistinguishable from one another. Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
“I inspect the set S and try to see whether it’s different from the set of all quarks, which we call Q. First I look at the cardinalities of S and Q. If their cardinalities were different, then obviously S and Q would be different sets. But their cardinalities are the same. Next I look for a quark that is contained in Q, but not contained in S. If there were such an element, then obviously S and Q would be different sets. But in order to successfully find such an element, I would have to make use of the distinctions between quarks. After all, how would I know that a given quark was in Q, but not in S? I would have to show that the quark in Q was distinct from each quark in S, but I have agreed to regard all quarks as indistinguishable. Therefore my search for an element of Q that is not in S will fail. I conclude that the set S is the same as the set Q. That is the set you gave me must be the set of all quarks.
“But this conclusion is obviously wrong. All I asked you for was an infinite set of quarks. There are many infinite sets of quarks, not all of which are the same as Q, the set of all quarks. You might have left some quarks out of S, and still provided me with an infinite set of quarks, which was all I asked for.
“Therefore we have a contradiction: I have proved something that is not necessarily true. Therefore the set of quarks cannot be infinite.”
--
The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets.
I take issue with your translation at only a single point:
My version contains a further constraint: When you ask me to bring you an infinite set of quarks, you instruct me to be as blind as you to the features that distinguish between quarks.
The_Duck tells metaphysicist to gather together an infinite set of quarks while remaining blind to their individuality. Metaphysicist, having no distinctions on which to carve infinite subsets, can respond to this request in only one way; include every quark. (I want to resist calling this the “set of all quarks,” because the incoherence of that concept with infinite quarks is what I argue.) The_Duck then goes out and finds another quark, and scolds metaphysicist, “You missed one.”
The_Duck is unjustified in criticizing metaphysicist, who must have picked “all the quarks,” given that metaphysicist succeeded—without knowing of any proper subsets—in assembling an infinite set . Having “selected all the quarks” doesn’t preclude finding another when they’re infinite in number and the only criterion for success is the number.
You will say that there is a fact of the matter as to whether the first set I assembled was all the quarks. Unblind yourself to the quarks’ individuating features, you say, and you get an underlying reality where the sets are different. I agree, but I think a more limited point suffices. When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark. There’s no way for me to distinguish the two sets. The contradiction is that following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
I’m making progress then. :)
No. If what you gathered is a proper subset of what you could have gathered, then you didn’t gather all the quarks, and you’re not justified in claiming that you did. How did you decide to leave out that one other quark? You must have made a distinction between it and the others that you did gather.
Of course there is. The superset contains a quark that the subset doesn’t. If you refuse to notice the differences that single that quark out from the others, that’s your loss.
I think that maybe you’re trying not to distinguish between quarks, but are implicitly distinguishing between “quarks that you know about” and “quarks that you don’t know about.” So you might assemble all the quarks you know about—an infinite number—and not have any evidence that this isn’t all the quarks there are. But later, you worry, you might find some other quarks that you didn’t know about before, so that your original set didn’t actually contain all quarks. This is not contradictory. If there was a chance that there existed quarks you didn’t know about, then you weren’t justified in saying that you had gathered all the quarks.
It does. If you’re not at the top of the hierarchy, you haven’t gathered all the quarks. And you can’t justify claiming that you’re at the top of the hierarchy by blinding yourself to evidence that would prove otherwise.