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ArisKatsaris comments on Natural Laws Are Descriptions, not Rules - Less Wrong
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I don't know why you retracted this, but I mostly agree with your comment. Tegmark IV and the ontological argument for god are, if not identical, at least closely enough related that anyone accepting the one and not the other should at least pause and consider carefully what exactly the differences are, and why exactly these differences are crucial for them...
I retracted it because when I wrote it I hadn't known Tegmarkism was part of Yudkowskian eclecticism. In that light, it deserves a less flippant response. While it strikes me as being as absurd as the ontological argument, for some of the same reasons, I can dispositively refute the ontological argument; so if they're really the same, I ought to be able to offer a simple, dispositive refutation of Tegmark. I think that's possible to, but it's instructive that the refutation isn't one that applies to the ontological argument. So, contrary to what I said, they're not really the same argument. Arguably, even, I committed what Yvain (mistakenly) considers a widespread fallacy, his "worst error," since I submerged Tegmark in the general disreputability of inference from possibility to necessity.
Briefly, Tegmark's analysis is obfuscatory because:
A. The best (most naturalistic) analyses of knowledge hold that it results from our reliable causal interactions with its objects. Thus, if Tegmark universes exist, we could have no knowledge of them (which leaves us with no reason to think they do exist).
I don't know how Tegmark addresses this objection. Or even if he does, but this objection seems to me the basic reason Tegmark's constructs seem so dismissible.
B. It's easy to "solve" many metaphysical and cosmological problems by positing an infinite number of entities, whether parallel universes or an infinite cosmos, but the concept of an actually realized infinity is incoherent.
[Side question: Does anyone happen to know whether the many-worlds interpretation of q.m. posits infinitely many worlds--or only a very, very large number?]
It's simpler to postulate that all possible worlds exist, rather than just one of them. Also, postulating an ensemble can be predictive, if you add the further postulate that you are a "typical observer in a typical world".
Panactualists need to hear the protests of more practical-minded people, to occasionally remind them that they really don't know whether the other worlds exist. The doctrine is either unprovable, undisprovable, or can be decided by a sort of insight we don't presently possess, such as one that can tell us why there is something rather than nothing.
No, it's not. Maybe it blows your mind to imagine space stretching away without limit, but if space is there independent of you, and if it has no edge, and if it doesn't close back on itself, then it's an actually realized infinity.
The second independent clause is true, but if (as I contend) actually realized infinities are incoherent, the proper conclusion is that the three assumptions cannot all hold.
Of course, having one's mind blown doesn't prove the concept entertained in incoherent; I must demonstrate that the concept really contains a logical contradiction. The contradiction in actual infinity is revealed by a question such as this one:
Assume there are an infinite number of quarks in the universe. Then, are there any quarks that aren't contained in the set of all the quarks in the universe?
Suggestion: Answer the question thoughtfully for yourself before proceeding to my answer.
By definition, they're all in the set. But, you can add a finite number to an infinite set and not change the number of elements. So, there are at the same time other quarks than are contained in the set of all quarks.
(I accept that Cantor demonstrated that infinities are consistent. The incoherence doesn't lie in the mathematics of infinity but in conceiving of them as actually realized. This was also the stance of mathematician and philosopher of mathematics David Hilbert, who devised the Hilbert's Hotel thought experiment to bring out the absurdity of actually realized infinities--while warmly welcoming Cantor's achievements in infinity taken strictly mathematically. Or as we might say, infinity as a limit rather than as a number).
Important changes for clarity Sept. 2.
Could you clarify this inference, please? How does the second sentence follow from the first?
Here's my interpretation of what you're saying: Let the set of all quarks be Q, and assume Q has infinite elements. Now pick a particular quark, let's call it Bob, and remove it from the set Q. Call the new set thus formed Q\Bob. Now, it's true that Q\Bob has the same number of elements as Q. But your claim seems to be stronger, that Q\Bob is in fact the same set as Q. If that is the case, then Q\Bob both is and isn't the set of all quarks and we have a contradiction. But why should I believe Q\Bob is identical to Q?
I agree that belief in the existence of actually infinite sets leads to all sorts of very counterintuitive scenarios, and perhaps that is adequate reason to be an infinite set atheist like Eliezer (although I'm unconvinced). But it does not lead to explicit contradiction, as you seem to be claiming.
Because there is no difference between Q and Q/Bob besides that Q/Bob contains Bob, a difference I'm trying to bracket: distinctions between individual quarks.
Instead of quarks, speak of points in Platonic heaven. Say there are infinitely many of them, and they have no defining individuality. The set Platonic points and the set of Platonic points points plus one are different sets: they contain different elements. Yet, in contradiction, they are the same set: there is no way to distinguish them.
Platonic points are potentially problematic in a way quarks aren't. (For one thing, they don't really exist.) But they bring out what I regard as the contradiction in actually realized infinite sets: infinite sets can sometimes be distinguished only by their cardinality, and then sets that are different (because they are formed by adding or subtracting elements) are the same (because they subsequently aren't distinguishable).
If Q genuinely has infinite cardinality, then its members cannot all be equal to one another. If you take, at random, any two purportedly distinct members of Q u and w, then it has to be the case that u is not equal to w. If the members were all equal to each other, then Q would have cardinality 1. So the members of Q have to be distinguishable in at least this sense -- there needs to be enough distinguishability so that the set genuinely has cardinality infinity. If you can actually build an infinite set of quarks or Platonic points, it cannot be the case that any arbitrary quark (or point) is identical to any other. If one accepts the principle of identity of indistinguishable, then it follows that quarks or points must be distinguishable (since they can be non-identical). But you need not accept this principle; you just need to agree with me that the members of the set Q cannot all be identical to one another.
Now, the criterion for identity of two sets A and B is that any z is a member of A if and only if it is a member of B. In other words, take any member of A, say z. If A = B you have to be able to find some member of B that is identical to z. But this is not true of the sets Q and Q\Bob. There is at least one member of Q which is not identical to any member of Q\Bob -- the member that was removed when constructing Q\Bob (which, remember, is not identical to any other member of Q). So Q is not identical to Q\Bob. There is no separate criterion for the identity of sets which leads to the conclusion that Q is identical to Q\Bob, so we do not have a contradiction.
Believe me, if there was an obvious contradiction in Zermelo-Fraenkel set theory (which includes an axiom of infinity), mathematicians would have noticed it by now.
I accept the principle, but I think it isn't relevant to this part of the problem. I can best elaborate by first dealing with another point.
True, but my claim is that there is a separate criterion for identity for actually realized sets. It arises exactly from the principle of the identity of indistinguishables. Q and Q/Bob are indistinguishable when the elements are indistinguishable; they should be distinguishable despite the elements being indistinguishable.
What justifies "suspending" the identity of indistinguishables when you talk about elements is that it's legitimate to talk about a set of things you consider metaphysically impossible. It's legitimate to talk about a set of Platonic points, none distinguishable from another except in being different from one another. We can easily conceive (but not picture) a set of 10 Platonic points, where selecting Bob doesn't differ from selecting Sam, but taking Bob and Sam differs from taking just Bob or just Sam. So, the identity of indistinguishables shouldn't apply to the elements of a set, where we must represent various metaphysical views. But if you accept the identity of indistinguishables, an infinite set containing Bob where Bob isn't distinguishable from Sam or Bill is identical to an infinite set without Bob.
I'll take your word on that, but I don't think it's relevant here. I think this is an argument in metaphysics rather than in mathematics. It deals in the implications of "actual realization." (Metaphysical issues, I think, are about coherence, just not mathematical coherence; the contradictions are conceptual rather than mathematical.) I don't think "actual realization" is a mathematical concept; otherwise--to return full circle--mathematics could decide whether Tegmark's right.
Among metaphysicians, infinity has gotten a free ride, the reason seeming to be that once you accept there's a consistent mathematical concept of infinity, the question of whether there are any actually realized infinities seems empirical.
Let me restate it, as my language contained miscues, such as "adding" elements to the set. Restated:
If there are infinitely many quarks in the universe, then I can form an infinite set of quarks. 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. But that set does not include all the quarks in the universe because finding other quarks is consistent with the set's defining [added 9/02] requirement that it contain infinitely many elements.
Could you (or anyone else) possibly provide me with a clue as to how I might find E.Y.'s opinions on this subject or on what you base that he's an infinite set atheist?
I'm also interested in how E.Y. avoids infinite sets when endorsing Tegmarkism or even the Many Worlds Interpretation of q.m. [In another thread, one poster explained that "worlds" are not ontologically basic in MWI, but I wonder if that's correct for realist versions (as opposed to Hawking-style fictitional worlds).]
If intuitions have any relevance to discussions of the metaphysics of infinity, I think they would have to be intuitions of incoherence: incomplete glimmerings of explicit contradiction. The contradiction that seems to lurk in actually realized infinities is between the implications of absence of limit provided by infinity and the implications of limit implied by its realization.
I'm still confused by this argument. Are you arguing in the second sentence that "any infinite set of quarks must be the set of all quarks"? But for example I could form the set of all up quarks, which is an infinite set of quarks, yet does not include any down quarks, and so is not the set of all quarks.
Are you implicitly using the following idea? "Suppose A and B are two sets of the same cardinality. Then A cannot be a proper subset of B." This is true for finite sets but false for infinite sets: the set of even integers has the same cardinality as the set of all integers, but the even integers are a proper subset of the set of all integers.
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.
Well, "site:lesswrong.com 'infinite set atheist'" is a clue, but http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/hkd is also a place to start.
No, then there are the same number of quarks in both cases in the sense of cardinality. Your intuition just isn't very good for handling how infinite sets behave- adding more to the an infinite set in some sense doesn't necessarily make it larger. Failure at having a good intuition for such things shouldn't be surprising; we didn't evolve to handle infinite sets.
Yes, I understand that; in fact, it was my express premise: "You can always add a finite number to an infinite set and not change the number of elements." That is, not change the number of quarks from one case to another.
Please read it again more carefully. My argument may be wrong, but it's really not that naive.
Added.
I see what you might be responding to: "So, there are more quarks than are contained in the set of all quarks." The second sentence, not the first. It's stated imprecisely. It should read, "So, there are other quarks than are contained in the set of all quarks." Now changed in the original.
You've collapsed the distinction between two possible worlds. You started out by saying, consider a universe containing infinitely many quarks. Then you say, consider a universe which has all the quarks from the first universe, plus a finite number of extra quarks. The set of all quarks in the second scenario indeed contains quarks that aren't in the set of all quarks in the first scenario, but that's not a contradiction.
It's like saying: Consider the possible world where Dick Cheney ended up as president for the last two years of Bush's second term. Then that would mean that there was a president who wasn't an element of the set of all presidents.
Replying separately to this now added comment. it still seems like this is an issue of ambiguous language. It isn't that there are other quarks that aren't contained in the set of all quarks." Is is that there's a set of quarks and a superset that have the same cardinality.
The problem seems to be that you are using the word "more" in a vague way that reflects more intuition than mathematical precision.
I think you responded before my correction, where I came to the same conclusion that my use of "more" was imprecise.
Added
I remember reading an essay maybe five years ago by Eliezer Yudkowsky where he maintained that the early Greek thinkers had been right to reject actual infinities for logical reasons. I can't find the essay. Has it been recanted? Is it a mere figment of my imagination? Does anyone recall this essay?
A "world" is not an ontologically fundamental concept in MWI. The fundamental thing is the wave function of the universe. We colloquially speak of "worlds" to refer to clumps of probability amplitude within the wave function.