A: But why are the dynamics of the electromagnetic field derived from Maxwell’s Lagrangian rather than some other equation? And why does the path integral method work at all?
B: BECAUSE IT IS THE LAW.
What do you think of Max Tegmark’s answer, that it’s because universes with every possible (i.e., non-contradictory) set of laws of physics exist and we happen to be in one with electromagnetic dynamics derived from Maxwell’s Lagrangian? (Or alternatively, every mathematical structure exist in a platonic sense and we happen to inhabit one that looks like this from the inside.)
I’m not sure if this can be called a LW consensus, but it has at least a large minority following here. One important reason is that this view seems to make it much easier to do decision theory, because it means that goals/values can be stated in terms of preferences about how mathematical structures turn out or unfold, instead of about “physical stuff”. In particular, UDT was heavily influenced by Tegmark’s ideas and there seems to be a consensus among people interested in decision theory here that UDT is a step in the right direction. If you’re not already familiar with Tegmark’s ideas, user ata wrote a post that can serve as an introduction.
This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise. I’ll try to summarize the context in which the idea of mathematical universe looks to me this way.
When abstract objects or ideas are thought about with mathematical precision, it turns out that they are best described by their “structure”, which is a collection of properties that these things have (like commutativity of multiplication on a complex plane or connectedness of a sphere), rather than some kind of “reductionistic” recipe for assembling them. These properties imply other properties, and in many interesting cases, even based on a fixed initial definition it’s possible to explore them in many possible ways, there is no restriction to a single direction in finding more properties (like new laws of number theory or geometry, as opposed to running a computer program to completion). At the same time, it’s not possible, either in principle or in practice, to infer all interesting properties following from given defining properties that specify a sufficiently complicated structure, so there is perpetual logical uncertainty.
When two structures (or two “things” having these respective structures, described by them to some extent) share some of their properties in some sense, it’s possible to infer new facts about one of the structures by observing the other. This way, for example, a computer program can reason about an infinite structure: if we know that a certain property stands or falls for the program and for the structure together, we can conclude that the property holds for the structure if its counterpart does for the program and so on. Also, setting up a structure that reflects properties of another one doesn’t require knowing all defining properties of that structure, knowing only sufficiently accurate approximations to some of them may be sufficient to make useful inferences.
Physical world then can be seen as just another thing that, to the extent it can be rigorously thought about, is described by certain properties or principles, of which we know only some and not precisely. Thinking about the world involves setting up certain things (brains, computers, experimental apparatus, abstract structures, physical theories, etc.) that capture some of its structure (these act as “maps” of the world), and then inferring more properties (making “predictions”) based on what they’ve managed to capture.
It doesn’t seem like there is much more to say on this big picture level, and treating physical world the same way we treat other complicated things, such as sufficiently complicated mathematical structures, seems like a natural thing to do. Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful. The physical world has structure, just as arithmetic has structure, but it doesn’t seem like much more can be said on this level of description.
Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful.
Every existing thing has a structure, but it is not clear that every logically consistent structure is the structure of an existing thing. The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless. The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
A way to tell an instantiated mathematical-structure-containing-sentient-beings from an uninstantiated one. (That doesn’t sound very different from telling conscious beings from philosophical zombies to me.)
I don’t know whether the concept of existence is meaningful. If it is, then something like the following should work:
To determine whether a mathematical structure M is instantiated, examine every thing that exists. If M is the structure of something that you examine, then M is instantiated. Otherwise, M is not instantiated.
Thus, whether the concept of existence is meaningful is the heart of the problem. I don’t claim to know that this concept is meaningful. I claim only not to know that it is meaningless.
I think it’s more like there are several concepts which share the same label. If a tree falls in the forest, and no one hears it, does it make a sound?
The tree in the forest is a case of various clear concepts (of sound) clearly implying different true answers.
The problem of Being is a problem of finding a clear concept that implies answers that many people find intuitively plausible.
It is more like the problem of being perfectly confident that various mathematical statements are true, while finding it very difficult to say just what it is that those statements are true about.
*points at objects which are instances of a class*
Those are instantiated (classes).
*points at classes that are unused at runtime, do not have any real object instances, perhaps were never even coded, but are simply logically consistent*
The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless.
The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
I’m making a distinction between saying “physical world is a structure” and “physical world has structure”: the first form seems to demand something unclear, and the latter seems to suffice for all purposes. Suppose things may either exist or not; but structure of things is abstract math, so it does seem clear that the properties of a structure don’t care whether it’s “instantiated” or not: the math works out according to what the structure is, regardless of which things have it. And since we only reason about things in terms of their structure, a distinction that isn’t reflected in that structure can’t enter into our reasoning about them.
(It might be possible to cash out “existence” of the kind physical world has as a certain property of structures, probably something very non-fundamental, like human morality, but this interpretation seems unlike the kind of confusion the argument is meant to counter.)
This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise.
I can’t find anything to disagree with after this quoted sentence, but “this seems like a trivial idea” certainly isn’t something I’d say if someone else wrote the comment you’re replying to. My guess is that you think “makes decision theory much easier” gives Tegmark too much credit because decision theory is far from solved, there are lots of hard problems left, and Tegmark’s ideas represent only a small step, in a relative sense, compared to the overall difficulty of the project.
If my guess is right, I could offer the defense that it feels like a large amount of progress to me, in an absolute sense, but it might be a good idea to just rephrase that sentence to avoid giving the wrong impression. Or, let me know if I’m totally off base and you intended a different point entirely.
I mean only that the description I sketched (which might be seen as referring the the idea of “mathematical universe”, but also deconstructs some of it, suggesting that it’s meaningless to insist that something “is a mathematical structure”), isn’t saying much of anything, and uses only standard ideas from mathematics; in this sense the idea of “mathematical universe” doesn’t say much of anything either (i.e. is trivial).
It might be a useful point to the extent that understanding it would banish useless ways of metaphysical theorizing about the physical world and free up time for more fruitful activities. So, my comment is unrelated to your point about decision theory, although the simplification (back to triviality) may be useful there and probably more relevant than for most other problems.
I’m aware of Tegmark’s ideas, although I haven’t thought about them much. I was not aware that they have a following on this site, probably because I haven’t read much of the decision theory material on here. I’ll read up on the idea and think about it more. My immediate uninformed inclination is skepticism, mainly on the grounds that I doubt the anthropics will work out in Tegmark’s favor without some gerrymandering of the ensemble. Also, being able to conceive of a mathematical structure as an independently existing entity rather than a formal description of the structure of some material system seems to require a Gestalt switch that I haven’t yet been able to attain.
What do you think of Max Tegmark’s answer, that it’s because universes with every possible (i.e., non-contradictory) set of laws of physics exist and we happen to be in one with electromagnetic dynamics derived from Maxwell’s Lagrangian?
I think Tegmark’s idea is either tautological or preposterous, depending on what he means by exist. If exist means ‘exist in an abstract, mathematical sense’ (as it does in the sentence There exist infinitely many prime numbers) then it’s tautological, and if it means ‘physically exist in this particular universe (i.e., the set of everything that can interact or have interacted with us, or interact or have interacted with something that can interact or have interacted with us, etc.)’ (as it does in the sentence Santa does not exist), it’s preposterous. The last chapter in Good and Real by Gary Drescher elaborates on this.
Once again, we badly need different words for ‘be mathematically possible’ and ‘be part of this universe’.
I assume he means they exist in the same sense as observers can only find themselves in places that exist. Which does not require any possibility of interaction between any two things that happen to exist.
I think Tegmark’s idea is either tautological or preposterous, depending on what he means by exist.
From what I can tell, Tegmark doesn’t mean either of the options you provide. It is closer to the first option (‘exist in the abstract’) but without all the implied privilege for the universe that happens to have you in it. The difference seems significant.
I don’t know too much about Tegmark, but I’m pretty sure he doesn’t have your second meaning in mind.
That said, I’m not sure your first meaning is actually tautological, given that for Tegmark’s idea to be an answer as Wei_Dai suggests, whatever “exist” means it has to encompass the kind of thing that you are doing right now.
The idea that things which “exist in an abstract mathematical sense” can, solely by virtue of that, do what you’re doing right now is perhaps tautological, but if so the tautology is not one that most humans will readily recognize as one.
Yes, I was unintentionally implicitly assuming that this universe is a mathematical structure. (OTOH, ISTM that this is a somewhat standard assumption on LW, e.g. Solomonoff induction wouldn’t make that much sense without it.)
Escape the first underscore by putting a backslash before it. (Why does the MarkDown italics mark-up work even within words, anyway? I think the situations where someone would want to italicize only part of a word are far fewer than those where one would want to use a word with an underscore in the middle of it.)
Why does the MarkDown italics mark-up work even within words, anyway? I think the situations where someone would want to italicize only part of a word are far fewer than those where one would want to use a word with an underscore in the middle of it.
I would think a lot less of a language that introduced an arbitrary limitation on its syntax like that. Italics of parts of a word come up occasionally and bold letters of a word more frequently than that. The language arbitrarily deciding it doesn’t want to execute the formatting commands unless you do whole words the same would be irritating, confusing and inelegant.
And it makes the rare-but-still-occasionally-desired case doable without escaping into HTML (which is not possible in LW’s no-HTML subset of Markdown).
You’d only need, whenever you see an underscore, to check whether the previous character is whitespace (or punctuation, e.g. a left parenthesis). Arundelo’s point seems more valid to me (though you might allow to escape spaces, e.g. _n_\ th… but that’d be more complicated).
What do you think of Max Tegmark’s answer, that it’s because universes with every possible (i.e., non-contradictory) set of laws of physics exist
If I can be frank, this is insane. This is the ontological argument for god revisited. Possibility does not imply necessity, and to think it does means you can rationally posit entities by defining them: defining them into existence.
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?]
if Tegmark universes exist, we could have no knowledge of them (which leaves us with no reason to think they do exist).
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.
the concept of an actually realized infinity is incoherent
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.
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).
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.
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.
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?
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.
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)
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.
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
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.
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’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.
Could you clarify this inference, please? How does the second sentence follow from the first?
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.
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.
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.
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 more quarks can be found and still be consistent with the only requirement that there be infinitely many quarks.
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.
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.
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.
But, you can always add a finite number to an infinite set and not change the number of elements. So, there are more 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. I understand that was Hilbert’s position.)
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.
No, then there are the same number of quarks in both cases in the sense of cardinality.
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.
So, there are other quarks than are contained in the set of all quarks.
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.
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?
[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?]
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.
What do you think of Max Tegmark’s answer, that it’s because universes with every possible (i.e., non-contradictory) set of laws of physics exist and we happen to be in one with electromagnetic dynamics derived from Maxwell’s Lagrangian? (Or alternatively, every mathematical structure exist in a platonic sense and we happen to inhabit one that looks like this from the inside.)
I’m not sure if this can be called a LW consensus, but it has at least a large minority following here. One important reason is that this view seems to make it much easier to do decision theory, because it means that goals/values can be stated in terms of preferences about how mathematical structures turn out or unfold, instead of about “physical stuff”. In particular, UDT was heavily influenced by Tegmark’s ideas and there seems to be a consensus among people interested in decision theory here that UDT is a step in the right direction. If you’re not already familiar with Tegmark’s ideas, user ata wrote a post that can serve as an introduction.
This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise. I’ll try to summarize the context in which the idea of mathematical universe looks to me this way.
When abstract objects or ideas are thought about with mathematical precision, it turns out that they are best described by their “structure”, which is a collection of properties that these things have (like commutativity of multiplication on a complex plane or connectedness of a sphere), rather than some kind of “reductionistic” recipe for assembling them. These properties imply other properties, and in many interesting cases, even based on a fixed initial definition it’s possible to explore them in many possible ways, there is no restriction to a single direction in finding more properties (like new laws of number theory or geometry, as opposed to running a computer program to completion). At the same time, it’s not possible, either in principle or in practice, to infer all interesting properties following from given defining properties that specify a sufficiently complicated structure, so there is perpetual logical uncertainty.
When two structures (or two “things” having these respective structures, described by them to some extent) share some of their properties in some sense, it’s possible to infer new facts about one of the structures by observing the other. This way, for example, a computer program can reason about an infinite structure: if we know that a certain property stands or falls for the program and for the structure together, we can conclude that the property holds for the structure if its counterpart does for the program and so on. Also, setting up a structure that reflects properties of another one doesn’t require knowing all defining properties of that structure, knowing only sufficiently accurate approximations to some of them may be sufficient to make useful inferences.
Physical world then can be seen as just another thing that, to the extent it can be rigorously thought about, is described by certain properties or principles, of which we know only some and not precisely. Thinking about the world involves setting up certain things (brains, computers, experimental apparatus, abstract structures, physical theories, etc.) that capture some of its structure (these act as “maps” of the world), and then inferring more properties (making “predictions”) based on what they’ve managed to capture.
It doesn’t seem like there is much more to say on this big picture level, and treating physical world the same way we treat other complicated things, such as sufficiently complicated mathematical structures, seems like a natural thing to do. Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful. The physical world has structure, just as arithmetic has structure, but it doesn’t seem like much more can be said on this level of description.
Every existing thing has a structure, but it is not clear that every logically consistent structure is the structure of an existing thing. The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless. The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
Define instantiated.
What would constitute a definition for your purposes?
A way to tell an instantiated mathematical-structure-containing-sentient-beings from an uninstantiated one. (That doesn’t sound very different from telling conscious beings from philosophical zombies to me.)
I don’t know whether the concept of existence is meaningful. If it is, then something like the following should work:
To determine whether a mathematical structure M is instantiated, examine every thing that exists. If M is the structure of something that you examine, then M is instantiated. Otherwise, M is not instantiated.
Thus, whether the concept of existence is meaningful is the heart of the problem. I don’t claim to know that this concept is meaningful. I claim only not to know that it is meaningless.
I think it’s more like there are several concepts which share the same label. If a tree falls in the forest, and no one hears it, does it make a sound?
The tree in the forest is a case of various clear concepts (of sound) clearly implying different true answers.
The problem of Being is a problem of finding a clear concept that implies answers that many people find intuitively plausible.
It is more like the problem of being perfectly confident that various mathematical statements are true, while finding it very difficult to say just what it is that those statements are true about.
*points at objects which are instances of a class*
Those are instantiated (classes).
*points at classes that are unused at runtime, do not have any real object instances, perhaps were never even coded, but are simply logically consistent*
Those are uninstantiated (classes).
Perhaps that’ll help seeing it.
I’m making a distinction between saying “physical world is a structure” and “physical world has structure”: the first form seems to demand something unclear, and the latter seems to suffice for all purposes. Suppose things may either exist or not; but structure of things is abstract math, so it does seem clear that the properties of a structure don’t care whether it’s “instantiated” or not: the math works out according to what the structure is, regardless of which things have it. And since we only reason about things in terms of their structure, a distinction that isn’t reflected in that structure can’t enter into our reasoning about them.
(It might be possible to cash out “existence” of the kind physical world has as a certain property of structures, probably something very non-fundamental, like human morality, but this interpretation seems unlike the kind of confusion the argument is meant to counter.)
I can’t find anything to disagree with after this quoted sentence, but “this seems like a trivial idea” certainly isn’t something I’d say if someone else wrote the comment you’re replying to. My guess is that you think “makes decision theory much easier” gives Tegmark too much credit because decision theory is far from solved, there are lots of hard problems left, and Tegmark’s ideas represent only a small step, in a relative sense, compared to the overall difficulty of the project.
If my guess is right, I could offer the defense that it feels like a large amount of progress to me, in an absolute sense, but it might be a good idea to just rephrase that sentence to avoid giving the wrong impression. Or, let me know if I’m totally off base and you intended a different point entirely.
I mean only that the description I sketched (which might be seen as referring the the idea of “mathematical universe”, but also deconstructs some of it, suggesting that it’s meaningless to insist that something “is a mathematical structure”), isn’t saying much of anything, and uses only standard ideas from mathematics; in this sense the idea of “mathematical universe” doesn’t say much of anything either (i.e. is trivial).
It might be a useful point to the extent that understanding it would banish useless ways of metaphysical theorizing about the physical world and free up time for more fruitful activities. So, my comment is unrelated to your point about decision theory, although the simplification (back to triviality) may be useful there and probably more relevant than for most other problems.
I’m aware of Tegmark’s ideas, although I haven’t thought about them much. I was not aware that they have a following on this site, probably because I haven’t read much of the decision theory material on here. I’ll read up on the idea and think about it more. My immediate uninformed inclination is skepticism, mainly on the grounds that I doubt the anthropics will work out in Tegmark’s favor without some gerrymandering of the ensemble. Also, being able to conceive of a mathematical structure as an independently existing entity rather than a formal description of the structure of some material system seems to require a Gestalt switch that I haven’t yet been able to attain.
LW discussions of anthropics in Tegmark’s multiverse:
http://lesswrong.com/lw/1zx/addresses_in_the_multiverse/
http://lesswrong.com/lw/535/anthropics_in_a_tegmark_multiverse/
http://lesswrong.com/lw/572/the_absolute_selfselection_assumption/
If you look in the comments of these posts you’ll find more links to earlier discussions.
I think Tegmark’s idea is either tautological or preposterous, depending on what he means by exist. If exist means ‘exist in an abstract, mathematical sense’ (as it does in the sentence There exist infinitely many prime numbers) then it’s tautological, and if it means ‘physically exist in this particular universe (i.e., the set of everything that can interact or have interacted with us, or interact or have interacted with something that can interact or have interacted with us, etc.)’ (as it does in the sentence Santa does not exist), it’s preposterous. The last chapter in Good and Real by Gary Drescher elaborates on this.
Once again, we badly need different words for ‘be mathematically possible’ and ‘be part of this universe’.
I assume he means they exist in the same sense as observers can only find themselves in places that exist. Which does not require any possibility of interaction between any two things that happen to exist.
From what I can tell, Tegmark doesn’t mean either of the options you provide. It is closer to the first option (‘exist in the abstract’) but without all the implied privilege for the universe that happens to have you in it. The difference seems significant.
I don’t know too much about Tegmark, but I’m pretty sure he doesn’t have your second meaning in mind.
That said, I’m not sure your first meaning is actually tautological, given that for Tegmark’s idea to be an answer as Wei_Dai suggests, whatever “exist” means it has to encompass the kind of thing that you are doing right now.
The idea that things which “exist in an abstract mathematical sense” can, solely by virtue of that, do what you’re doing right now is perhaps tautological, but if so the tautology is not one that most humans will readily recognize as one.
Yes, I was unintentionally implicitly assuming that this universe is a mathematical structure. (OTOH, ISTM that this is a somewhat standard assumption on LW, e.g. Solomonoff induction wouldn’t make that much sense without it.)
Perhaps. But the connotations of saying that something exists in an abstract, mathematical sense tend to run counter to that.
Escape the first underscore by putting a backslash before it. (Why does the MarkDown italics mark-up work even within words, anyway? I think the situations where someone would want to italicize only part of a word are far fewer than those where one would want to use a word with an underscore in the middle of it.)
I would think a lot less of a language that introduced an arbitrary limitation on its syntax like that. Italics of parts of a word come up occasionally and bold letters of a word more frequently than that. The language arbitrarily deciding it doesn’t want to execute the formatting commands unless you do whole words the same would be irritating, confusing and inelegant.
It’s probably less work to read character-by-character than to split on words and read the first and last character of each.
And it makes the rare-but-still-occasionally-desired case doable without escaping into HTML (which is not possible in LW’s no-HTML subset of Markdown).
You’d only need, whenever you see an underscore, to check whether the previous character is whitespace (or punctuation, e.g. a left parenthesis). Arundelo’s point seems more valid to me (though you might allow to escape spaces, e.g.
_n_\ th… but that’d be more complicated).True! I do not know why MarkDown italics works within words.
If I can be frank, this is insane. This is the ontological argument for god revisited. Possibility does not imply necessity, and to think it does means you can rationally posit entities by defining them: defining them into existence.
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.
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.”
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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.
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.
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.