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First, I'd like to make an apology/correction about a couple things I said in part 1. A friend of mine pointed out to me that some of the things I said could be considered insulting to philosophers, in particular people working on problems in philosophy of mind. I did not intend my comments to be insulting, and I think some of what I said was phrased in an unfortunate way that made it sound more dissmissive than I'd intended, and than I have any justification for claiming. First, I want to make it clear that I have a great respect for the brilliant people who work on philosophy. It's not an easy subject, and what I was not saying was "wow, these questions are all so easy, why hasn't anyone solved them yet?" What I was instead trying to do was establish a comparison between most of the problems in philosophy (which I see as revolving around language--which is not an insult, but merely a statement of how I personally view philosophy, not too dissimilar from how Wittgenstein viewed it), and the "deep" problems I see in philosophy of mathematics regarding the foundations of mathematics. I consider those questions, in particular to be very hard. And when I said that other problems were "easy" I meant in comparison to those, not easy as in trivial, something anyone could do. I took some pot shots at philosophy of mind, but in truth I was focusing on my personal views regarding qualia, and I made the statements there a bit too broad. Also, none of what I said is immutable, it's just how I currently happen to feel about the subject. I'm sure there are many interesting problems in philosophy of mind, many of which I am unaware of or haven't even thought about. So for me to say "they're mostly solved" is a huge overstatement and should be narrowed down to refer just to the problems I was thinking about. At any rate, take anything I say in my journal with a grain of salt, as it is all an attempt at me representing my present thoughts and opinions, by no means an eternal statement of fact (I know that sometimes the way I word things, can make it sound that way.) With that out of the way, I present to you.... part 2:

I mentioned in part1 that I think the thing Copenhagenists reject about the physical world which Many-Worlders accept is a particular brand of reductionism. I want to explain what this sort of reductionism is (let's call it R1 to avoid confusion with other reductionist positions), and why it makes a difference.

R1 is essentially the assumption that the microscopic world is more fundamental than the macroscopic world, rather than the other way around. This is a premise that Bob Laughlin explicitly rejects, and I think it's a premise that you have to reject if you're a true Copenhagenist. Otherwise, as far as I can see, the position of Copenhagenists becomes somewhat untenable.

What do I mean by micro and macro and where is the dividing line? First, it does NOT mean large versus small distance scales. In quantum gravity, there is something called the UV/IR correspondence principle which may invert the roles of these in the ultimate theory. This makes the choice of words "micro" and "macro" somewhat unfortunate, but nevertheless it's traditionally the way quantum mechanics is discussed, and accurate enough for my purposes here, so I'll adhere to that convention. What I mean by it has to do instead with the number of degrees of freedom in a system. If there are a large number of interacting degrees of freedom, it's a macroscopic system (sometimes called a "classical" system) whereas if there are a small number of degrees of freedom it's a microscopic system. There is no unique place where you have to put the dividing line, which I think is one of the biggest problems with Copenhagen, but I do think there is a possibility that it doesn't matter that the dividing line is not uniquely defined. I'll get to that possibility later.

The rejection of R1 is not just something that arises in the Copenhagen interpretation of quantum mechanics. It's something that is expressed in a more general form by the same sorts of physicists when trying to make sense of how to connect statistical mechanics with thermodynamics. Often, people who work with or teach statmech will say things like "large numbers do not behave the same as small numbers. They have special properties!" This is very closely related to the rejection of R1, but just to be careful let's call the mathematical realist proposition they are rejecting here M1, which I'll explain in more detail later. The dividing line between micro and macro systems comes from the same dividing line that physicists who reject M1 imagine exists between large and small numbers. Both my advisor, and the condensed matter physicist I worked with earlier in grad school have expressed this opinion to me regarding numbers, and even stated it as fact. But I, as a reductionist, and a heretical mathematical realist, do not believe it. And when it comes down to it, it's really the reason why I subscribe to Many Worlds instead of Copenhagen. Nevertheless, I'm quite interested in exploring the consequences of rejecting R1 to see if this alternate worldview (Copenhagen + nonreductionism + mathematical formalism) is really consistent. If it is, then I would probably prefer it over Many Worlds; but I've just never been able to verify for myself that it's consistent; I always run into paradoxes when I try to make sense of it.

The basic view of those who reject M1 is that when you have very large numbers, on the order of N= e^(10^23) or more, it magically becomes true that N = N+1. The reason they believe this is because they are antirealists about mathematical objects such as integers, and they view mathematics only as a human construction, similar to language, whose purpose is to describe the physical world. Since it would take longer than the age of the universe to ever count up N things one by one (where N is still the number above), and verify that N is perhaps not really the same thing as N+1, it is meaningless they say, to distinguish N from N+1 since they refer to two things which only differ by an unmeasurable and unphysical property. According to them, the property which they alledgedly differ by, does not exist since it cannot be verified through the senses or even with any sort of known instrumentation which extends the senses. (Again, you can see the influence of the logical positivists on this way of thinking.)

But wait, you say, isn't the statement that N is equal to N+1 absurd? How can they believe that is true of any number? No number has that property! Not so fast, their position cannot be dispensed with so easily. There are, in fact, numbers where N = N+1. However in my worldview, this is only ever true of infinite numbers. It is not true of any finite number N no matter how large. Furthermore, I believe that even though you cannot measure the difference, there IS a difference between infinite numbers and very very large numbers. They are similar in many respects, but strictly speaking they are not the same thing. The people who reject M1 would say this belief of mine is unjustified and faith-based, since I cannot measure any difference with a measuring instrument. The only way to prove there is a difference is to apply the axioms of mathematics, which--according to the mathematical formalists who reject M1--is just a formal system which doesn't actually represent any real things called "numbers". To the mathematical formalist, the only numbers which exist are those which can be used to actually represent something in the physical world. If they can't be used to represent something in the physical world then any proofs about them are strictly formal language games, not actually statements that refer to reality. And the key to understanding how their position makes sense is that large numbers like the N I listed above can be used to represent reality (for instance, it can be used to represent the number of microstates corresponding to a single macrostate in statistical mechanics)--it's just that they're large enough that N and N+1 both represent the same physical state as far as anyone can measure. Therefore, they do not believe there is a difference between them. So far, there is nothing inconsistent about the things that people who reject M1 and R1 say. There is only the dogma that people like Plato, Godel, Penrose, myself, and other mathematical realists share that those large but finite numbers *do* really exist, even though humans could never count up to them, even with drastically improved neural firing rates. (Incidentally, I hate to lump myself in with Plato, Godel, and Penrose, because I think they were all crazy in their own way, and allowed their mathematical realist beliefs to spill over into anti-empiricism. However, on this point--the acceptance of M1--I agree with them. Later, I'll talk more about where I disagree with them.)

So fine, some people treat large numbers ontologically differently from small numbers. What's the big deal? What does any of this have to do with interpretting quantum mechanics? Well, in quantum mechanics the basic "measurement process" occurs when a system that is large in the sense described above interacts with a small system, entangling the state of the large "classical" system with the quantum state of the small system (which can be taken only to be a single degree of freedom-- one qubit--for simplicity). The disagreement between Copenhagenists and Many-Worlders is over whether it is meaningful to talk about the state of the qubit before it is measured at all. For if it is meaningful, quantum mechanics predicts that this measurement process will evolve the entire system into a state which is decomposable into two separate non-interacting pieces (worlds, universes, branches, whatever you want to call them): one where the classical system "measured" the qubit as being 1, and the other where the classical system "measured" it being 0. (In the case of the proverbial cat, you end up with one universe where the cat is dead, and one where it is alive.) It's worth mentioning that the only case where these two branches become perfectly distinct, perfectly separate universes, is where you have a system of size N, where N is equal to infinity. For a large but finite N, they are not completely separate in principle, however there is no way they can interact in any measurable way within a time frame humans could witness. Both of the universes will have long since died of heat death and no longer be able to support intelligent life, before they would ever interact with each other again. Copenhagenists agree with everything I've said so far in this paragraph. Where they disagree is that they then take the fact that humans will never be able to witness the interaction of the two universes again as grounds for rejecting the existence of whichever universe we do not end up finding ourselves in after the measurement. Actually, it's not even that they *disbelieve* in the existence of the other universe. It's that, like good little logical positivists, they do not believe it is a meaningful question to ask whether it exists. The more reasonable ones may say something like "well, it's just a matter of philosophy." But the tone in which they say it, sort of implies that philosophy is uninteresting and that they couldn't care less about questions like this. But the more hardcore ones--the true Copenhagenists--will insist that the question of whether the other universe exists or not is entirely meaningless, since we cannot verify it with our sensory perceptions.

So now I've gone back to talking about quantum mechanics, but I still haven't connected it up with R1 or M1. I'm getting there, hold your horses! Based on the paragraph above, it would appear the Copenhagenists have a consistent--if bizarre--way of looking at the world. However, it starts to seem less defensible (to me) as you start to ask questions about what kinds of things exist in the world. I think that most Copenhagenists would agree with me that large, macroscopic objects like baseballs and ponies, exist in an approximate sense even though they do not exist in the strict sense. (I've written extensively on this before, and it's the reason my journal is named spoonless--large objects such as spoons exist as concepts in our minds, which appoximately represent the things which are going on in the external world, however they are not perfectly separable from the non-spoonlike stuff out there). While I'm an antirealist about large macroscopic objects in the strict sense, and so are Copenhagenists I think, where we differ is in our opinions about microscopic objects. They would say that the approximate existence of macroscopic objects is an emergent property. [edit: after writing this paragraph, I realized it's possible that Copenhagenists just don't believe in any strict sense of exist like I do at all... maybe their only notion of "exist" is in the approximate sense. Which is pretty interesting.] Individual degrees of freedom, they believe, do not actually exist in any sense of the word--they are purely mental fictions that we've used to try to helps us predict how the macroscopic objects we are directly aware of interact and behave. This is the point where it really comes down to them rejecting R1. If they don't reject R1, I think their position is untenable. With rejecting R1, I'm still unsure but their position seems so bizarre and radical to me that I have difficulty making sense of it. To me, macroscopic objects only approximately exist, whereas the microscopic degrees of freedom--whatever they may turn out to be (even if they end up being "large" and non-local in terms of distance scales)--those are what actually exists in the strict sense of exist. This is the brand of reductionism that I subscribe to, and this is what I call R1.

Going further, trying to understand how Copenhagen + rejection of M1 + rejection of R1 *might* form a consistent picture of the world. Most of the time, while we live our lives, we don't excite degrees of freedom above a certain energy scale. We're only probing degrees of freedom at a certain scale, usually lower energy than even atomic scales. Only occasionally do chemical processes happen which change one molecule into another, and even more rarely do nuclear processes happen that change one atom into another. And only *extremely* rarely do we do experiments in particle accelerators, where we actually witness different subatomic particles changing into each other, and changing back and forth between matter and pure energy. So rolling history back before we built such large accelerators, or even before we'd witnessed nuclear processes directly, would it be consistent to believe that only the low energy world actually exists, since to a fairly good degree of accuracy, you can describe it without invoking all of these higher energy processes (which involve postulating more microscopic degrees of freedom which the macroscopic physics emerges from)? I think the answer is no, but sometimes I'm not sure. Perhaps you have to believe there is *some* microscopic physics going on, but you could consistently believe that it's meaningless to ask which sort of microscopic physics is going on, if several of them would give rise to the same macroscopic world we observe. Then the question is, what happens when you actually excite the microscopic degrees of freedom? Have you suddenly forced the universe to choose one particular implementation over all the others... and before that it was meaningless to ask which one was true? This seems very similar to the questions about what is the case before you open the box containing Shrodinger's cat. On the one hand, I think it is definitely a very misleading statement to say that it was "chosen" because of your observation. But on the other hand, I think there is a sense in which something not too unlike this could be true, and this is the sense that would need to be true in order for the Copenhagen Interpretation to be fully coherent. Unfortunately, this leads me back into a lot of long complicated discussion about philosophy of mathematics. Which I'm going to have to save for part 3. But just to give you a sneak preview of where I'm going with this, here's one more paragraph...

I think that the sense in which it could be true is this: it may be that you can only rigorously define an observer by specifying the entire history of what they have observed. In other words, there may be no matter of fact truth about which universe a particular observer is "living in"; observers which have identical perceptions which could be mathematically described by different fundamental physics should perhaps be labelled as the same observer. I already suspect this is true to some extent with respect to the different branches of the quantum multiverse. So this would be a more radical version of the same principle, applied instead to Max Tegmark's Level IV multiverse which includes all mathematically consistent structures. In other words, it could be that taking Many Worlds very seriously and then taking the Level IV multiverse seriously, and choosing a certain path regarding the language you use to talk about it, ends up leading you eventually back to Copenhagen. I still think it's unlikely and has certain problems, but I don't rule it out as a possibility and it would be extremely fascinating if it turns out to be true.

I suspect I will not have time to write part 3 and part 4 until after I get back from Christmas break. Nevertheless, I still have a lot more to write, in particular stuff about Godel's Incompleteness theorem, and philosophy of mathematics. I'm almost finished with explaining my thoughts on quantum mechanics.


( 16 comments — Leave a comment )
Dec. 11th, 2007 08:35 pm (UTC)
Don't take this the wrong way, but your paragraphs are intimidatingly long.
Dec. 11th, 2007 09:02 pm (UTC)
Very interesting stuff - I think I'll take that paragraph you wrote motivating ultrafinitism and re-post it on my journal. I think my math friends will have something interesting to say (it might just be more of "look how stupid physicists are" but many of them have positivist scruples as well, so it might be interesting).
Dec. 12th, 2007 01:15 am (UTC)
I'll keep my eyes peeled to see if they have some interesting comments on it.
Dec. 12th, 2007 12:22 am (UTC)
Fascinating! I wish I knew more about these topics...but in my view, just because some property of some number or physical system cannot be measured using our current instrumentation does not necessarily mean that it doesn't exist. Even if something is meaningless to ask about as our observation itself would affect what we see, or if it's not detectable in our observed universe, it might still serve as the underlying reason behind something we notice, or something we can't explain as of yet.

We might only be able to say for sure that we know of approximations of reality (i.e. n = n +1 for large n, low energy vibrations, etc) but that doesn't mean the approximations ARE reality.

As an analogy - Einsteinian physics could well have existed before we could observe anything other than classical Newtonian physical laws. It simply had effects we couldn't notice then.

I tend to go by Bertrand Russell's simplicity principle - choosing the simplest explanation for phenomena when faced with equally compelling explanations. However, that's just mainly for the sake of my sanity, not because I believe something is necessarily more true or more meaningful in the external world.

Thanks for contributing intellect to my friends-page!
Dec. 12th, 2007 01:20 am (UTC)
I'm pleased to see you seem to have understood some of the key points I was making. I was worried that my writings on this would be too obscure or technical for more than 3 or 4 people on my friendslist to get much out of it.

I'd definitely agree with what you say here.
Dec. 12th, 2007 09:53 am (UTC)
Some of Greg Egan's fiction has fun with these themes. I'm thinking particularly of Distress and Permutation City.
Dec. 13th, 2007 04:16 am (UTC)
Alright so you believe in the many-worlds interpretation of quantum mechanics. Could you please describe what according to you are its implications for (libertarian) free-will? Thanks.
Dec. 13th, 2007 12:08 pm (UTC)
Also, the many-worlds interpretation has the problem of being dualistic in nature, regarding consciousness as a primitive unexplained entity, quite different from the substance of the physical world. I think Penrose's Objective Reduction avoids the problems of both copenhagen and many-worlds interpretation. You may also like to have a look at section 6 of this: http://cms.brookes.ac.uk/staff/PeterElls/FreeWill/NaturalisticFreeWill.rtf
What say?
Dec. 13th, 2007 06:34 pm (UTC)
Many worlds is a strictly materialist framework, where consciousness is not at all a fundamental entity and plays no role whatsoever in the theory. It is simply an emergent property of matter, which behaves the same way as any other matter. Any sane interpretation of quantum mechanics has this property, as consciousness obviously has nothing to do with quantum mechanics (the two being at opposite ends of the complexity spectrum).

One of the biggest advantages of many worlds is that there is absolutely no need for a reference to an "observer" in order to formulate the theory. As far as I know, it's the only interpretation where this is the case... in most other interpretations, you need to talk about observers as being special in at least some sense (where the term "observer" means a large classical system, and has nothing to do with consciousness in the first place).

Edited at 2007-12-14 03:34 am (UTC)
Dec. 14th, 2007 05:08 am (UTC)
The Objective Reduction (or Objective Collapse) interpretation also doesn't make observers special in any sense. And the fact that Copenhagen has problems doesn't make many-worlds automatically true. In fact there are many serious problems with Many-worlds, which you are probably aware of, like the basis problem, and accounting for the statistical probabilities of quantum mechanics.
Dec. 14th, 2007 05:38 am (UTC)
I guess I wouldn't consider Penrose's theory an "interpretation", since it is based on speculative new physics beyond quantum mechanics.

The basis problem is, as far as I can tell, just a straw man. The Many Worlds Interpretation that I believe in certainly doesn't require any preferred basis. There may be some Many-Worlders who believe in a preferred basis as a fundamental part of the theory, but I think this is mostly just a straw man that non-Many-Worlders invented.

I agree that there are some remaining issues to be clarified regarding the derivation of probabilities from unitary evolution. There are fairly convincing arguments that it can be done, but some of the steps I think could be solidified more.
Dec. 14th, 2007 08:37 am (UTC)
Alright ... but you do agree that the many-worlds interpretation makes consciousness an epiphenomenon, correct?
Dec. 15th, 2007 03:17 am (UTC)
No, not at all. Consciousness is a physical process, like any other physical process. There is no need to invoke ephiphenomenal bullshit in order to explain it.

The only reason someone might believe that many-worlds leads to epiphenominalism is if they think materialism always leads to epiphenominalism. It does not. Materialism does not have any gaps or "hard problems" associated with it. Consciousness works just fine as an objective physical process, and has nothing to do with physics or quantum mechanics, aside from being in principle reducible to biology which is reducible to chemistry which is reducible to physics. That's really the only connection.
Dec. 15th, 2007 03:21 am (UTC)
By the way, why on earth did you think "The Religious Society of Friends (Quakers) in Britain" might have something non-idiotic to say about physics? Their claim that many worlds is "dualist" is complete nonsense.
Dec. 13th, 2007 12:16 pm (UTC)
Ok I read one of your earlier posts. Seems like you are compatiblist and agree with Dennett's views.
Dec. 13th, 2007 06:34 pm (UTC)
Yes. I don't see how any theory of quantum physics could possibly have implications for freewill. I've heard people claim that, but I think they are just being silly.
( 16 comments — Leave a comment )


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