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The_Duck comments on Natural Laws Are Descriptions, not Rules - Less Wrong
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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.
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"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."
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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.
TheDuck 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.) TheDuck 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.