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Tegmark's Mathematical Universe

  1. Nov 28, 2007 #1

    Chronos

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    Max Tegmark has interesting, if controversial ideas about cosmology. I read this paper today and was intrigued. Any comments?

    http://arxiv.org/abs/0704.0646
    The Mathematical Universe
    Authors: Max Tegmark (MIT)
    (Submitted on 5 Apr 2007 (v1), last revised 8 Oct 2007 (this version, v2))
    Abstract: I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.

    I reflexively dismiss this idea because it appear untestable, but, he makes a compelling argument this objection may be illusory. The symmetry breaking thing really intrigues me. The apparently broken symmetries could be restored in parallel universes. I find that idea strangely attractive.
     
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  3. Nov 28, 2007 #2

    marcus

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    Didn't you realize?
    Tegmark himself is an abstract mathematical structure.

    They used to hang out at Schraft's on Park Avenue and 53rd street, eating abstract mathematical icecream, but recently they have been duplicating themselves and gradually infiltrating Think Tanks and Major Universities.
     
  4. Nov 28, 2007 #3

    Chronos

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    Hilarious, marcus. His posits on the brain thing were amusing. His reasoning, however, was interesting. I think he made a case that Godel's 'incompleteness' conjecture is not bullet proof.
     
  5. Nov 28, 2007 #4
    I think Tegmark's idea is really interesting, but as I see it it is fundamentally not a scientific idea. It is a philosophical idea. I don't think we can ever use it to draw useful conclusions except in a philosophical sense.

    The one obvious, useful conclusion that I'm aware of which comes out of Tegmark's idea-- "we should expect the ultimate laws of our universe to have a description which is simple, in the k-information theory sense"-- one could perhaps reach in other ways without specifically positing Tegmark's whole mathematical multiverse.

    I'm a little confused where Godel comes in. I may try to read that paper later and see if I have anything to say on that particular point.
     
  6. Nov 28, 2007 #5

    Ben Niehoff

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    If the "true physical laws" (whatever they are) have a finite mathematical description, and these laws encompass all of the possible phenomena of reality, then the idea that reality is a mathematical abstraction is tautologically true (it would, after all, be isomorphic to a mathematical abstraction, which is good enough as far as any sensible philosophical definition of "is"). As such, this idea is both fascinating and meaningless at the same time.

    The other possibility is that reality admits no finite mathematical description, which means there are ultimately no fixed rules; but it does mean physicists won't be out of a job any time soon.

    In either case, this is the stuff of philosophy, not physics.
     
  7. Nov 28, 2007 #6

    Fra

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    Perhaps I misunderstand what he means but I skimmed this paper before and when I read the paper my impression is that his constructs and the ERH somehow has the purpose of justifying a reductionist approach.

    My main objection is that the birds view as he calls it, might not fit into a frog projection due to information capacity limits. So doesn't that miss see point?

    He seems to look for a bird view, so that he can explain the frog view? But I consider myself the frog in this context, not the bird. I can "try to be a bird" but my nature probabyl limits me?

    Imagine that the the "size of the view" is constrained by the information capacity of the observer. Certainly the environment can be used to extent the information storage, but then it can hardly be immediately available because we ran into the issue of information content vs compression vs processing. And if we reduce this into "computing" and "decoding", that is fine, but then what's the different from normal "learning"?

    I like parts of the paper, but I don't think I share his apparent philosophy of science.

    Edit: I think I said this in another thread, but relative to say an atom in a lab, a human + the lab could probably fairly well satisfy the requirements for a birds view, informationwise. But relative to a remote object in space, or on the cosmological scale I am nothing but a tiny frog.

    /Fredrik
     
    Last edited: Nov 28, 2007
  8. Nov 28, 2007 #7
    That seems like borderline philosophy to me :bugeye:
     
  9. Nov 28, 2007 #8

    Chronos

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    Philosophy relies on logic, an often useful instrument in the 'tool box' of science. Mathematics is merely a more rigorous extension of logic. I'm pretty firmly attached to the idea the universe must be mathematically consistent. I perceive Tegmark has framed a persuasive argument to this effect. But, I'm also attached to information theory as a powerful tool in understanding the universe.
     
  10. Nov 29, 2007 #9

    Fra

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    For example, tegemarks seems to distinguish between apparently random and "true random".

    In terms of information exchange - what is the difference until the "algorithm is cracked"?

    How _long time_ does it take to "crack an algorithm"? In the general case?

    The question is then, how can a definition be made _in time_, that doesn't incorporate a specualtion about future findings? If it can't be made, then again - what's the difference?

    /Fredrik
     
  11. Nov 29, 2007 #10

    marcus

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    Borderline seems right. But could it be borderline something else?
     
  12. Nov 29, 2007 #11

    marcus

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    Seriously, I agree with Coin's comment, at least the second part:

    Chronos mentioned this in the original post. Untestability even in principle, the implication being that it doesnt fall within the domain of empirical science.

    Probably there should be a thread discussing Tegmark mathematical universe idea in the Philosophy forum. Maybe there is one already. Did anyone check?

    edit: I checked in Philosophy
    https://www.physicsforums.com/forumdisplay.php?f=112
    and they don't have a Tegmark thread, at least so far.
     
    Last edited: Nov 29, 2007
  13. Nov 29, 2007 #12

    Fra

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    A general forum question: Maybe I haven't seen the feature yet, but is there somewhere a "list of all posts (in any section)" since the last time visiting the forum?

    Becase it could be that an interesting post is put under an "odd topic". For example, I would not a priori expect physics related dicussions under philosophy - which means I don't even look there. Philosophy is such a wide topic ranging almost from psychology to science. IMO, philosophy is not a sensible topic on it's own. Rather one usually refers to something more specific, for example the _philosophy of science_. IMO someone who thinks the _philosophy of science_ (not philosophy in general) is irrelevant to science should think again ;)

    This probably has the effect that one has to place the post, where it's most likely to be read by the intended group of readers, which gets wrong? If there was such a think as a list of new posts (listing the subjects) it would be easier to find interesting things regardless of where it's posted?

    Is there such a feature on here that I missed? Anyone else missing this?

    /Fredrik
     
  14. Nov 29, 2007 #13

    Fra

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    I guess I didn't look very hard because I thought it wasn't there :) I found the feature.

    /Fredrik
     
    Last edited: Nov 29, 2007
  15. Nov 29, 2007 #14
    Indeed, we are all programs/algorithms run by a brain, therefore we are universes in our own right that happen to be simulated in this universe (described by the Standard Model + GR). This idea provides us with a better way to think about a whole host of philosophical ideas and bring them back into the realm of physics/mathematics.

    E.g. what are qualia (i.e. if I see the color blue, then what is this color blue I'm seeing)? What else could they be than events in the universe simulated by the brain?

    Tegmark writes in his articles about his idea that the fundamental problem is to come up with a measure over the space of all mathematical models. I think that it is unavoidable that the measure is some function of the Kolmogorov complexity of the model such that more complex models are assigned a low measure.

    This then leads to extremely complex universes like us that need trillions of arbitrary parameters to define finding themselves being simulated in simple universes instead of finding themself in their own universe.
     
  16. Nov 29, 2007 #15
    IMO the MUH is not nearly as important and vital as the CUH- this is a fundamental issue of the ontology of existence- what existence is and what can be

    I said in another thread: I am a proponent of something like Max Tegmark's http://arxiv.org/abs/0704.0646" [Broken] which states that Reality is defined by computable causal systems and that only computable structures 'exist'- or more precisely computability IS existence- since only causal structures with logical/consistent/computable relations and rules can provide the medium of isotropic space and time for which a world and it's history could be physically expressed as real-

     
    Last edited by a moderator: May 3, 2017
  17. Nov 29, 2007 #16

    Chronos

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    Agreed, setAI. The CUH is the more fundamental issue. A causal universe [which ours appears to be] must yield logically consistent experimental results. It is otherwise difficult to imagine how our univere has persisted for so long. This concept may not be inherently predictive, but has utility in eliminating untenable solutions.
     
  18. Nov 30, 2007 #17

    Fra

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    I agree completely with this.

    The way I envisions this, the reason for this constraint is that the probabilities are conditional and they must be formulated in terms of something the obserever can relate to. And if we think that the encoding capacity of an observer is generally not infinite this implies two things:

    1) Overly complex things are automatically assigned a low measure, and this is an explicit realisation of occams razor or thta simplicivy is a virtue. It ultimately boils down to fitness. A overly complex model is not as fit because it adapts poorly.

    2) The limiting relational capcity of the observer implies that there are bound to be limits to what he can predict. He knows what he knows, but it's not possible to assign bounds on what you don't know. However, there is also no reason to make assumptions on what you don't know. So as long as there is no negative feedback due to this ignorance forcing the observer to remodel, the ignorance of the observer could be stable.

    This is a fundamental key in my own thinking.

    /Fredrik
     
  19. Dec 3, 2007 #18

    Chronos

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    Just a side note - the paper was submitted to 'GR-QC' [of Arxiv], not 'physics.hist-ph'.
     
    Last edited: Dec 3, 2007
  20. Mar 6, 2008 #19
    I have compiled a list of my remarks on this MUH hypothesis and the related CUH at sci.logic.

    There are 6 posts of mine there right now that have not been responded to, perhaps because they are logicians (and perhaps because I posted this recently) even though there is a fair amount of logic in my posts.

    Should I post a link to that, copy and paste those six posts here, or what? Does it matter?
    I'll start with the link, since I don't think that will consume this board's hard drive space:

    http://groups.google.sh/group/sci.l...baa707749ad/ef7752e4bcfc2631#ef7752e4bcfc2631
     
  21. Mar 6, 2008 #20

    In reference to the link I just posted, I believe I made some remarks at that link that could be relevant to this line of thought.
    Posted Mar 4, 3:06pm
     
    Last edited by a moderator: Apr 23, 2017
  22. Mar 6, 2008 #21
    There is a recent paper arguing that consciousness is inherently non-computable, which would seem to falsify the CUH.

    Non-Computability of Consciousness
    http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.1617v1.pdf

    Abstract:
    With the great success in simulating many intelligent behaviors using
    computing devices, there has been an ongoing debate whether all
    conscious activities are computational processes. In this paper, the answer to
    this question is shown to be no. A certain phenomenon of consciousness is demonstrated to be fully represented as a computational process using a
    quantum computer. Based on the computability criterion discussed with Turing machines, the model constructed is shown to necessarily involve a
    non-computable element. The concept that this is solely a quantum effect
    and does not work for a classical case is also discussed.




    However, Tegmark has a rejoinder that apparently amuses some people, rather than persuades them?
    http://space.mit.edu/home/tegmark/brain.html



    Tegmark makes a case for brain function being modeled adequately with
    classical theoretical means (possibly such as Turing machines) and
    that brains do not function like quantum computers. (Essentially the
    main factor is that the brain is not nearly at absolute zero degrees,
    or otherwise in an environment in which superposition type effects
    that consciousness apparently mimics well enough to keep many on the
    fence, is more common than Earthly temperatures where our brains
    normally reside.)

    If Tegmark does prove his point, while others in his community remain
    skeptical that brain function is +not+ an example of a quantum
    computer, then the paper I cited about the non-computability of
    consciousness does not invalidate Tegmark's CUH, mentioned in section
    VII of the first link in http://groups.google.sh/group/sci.l...aa707749ad/ef7752e4bcfc2631#ef7752e4bcfc2631". The non-computability of
    consciousness would seem to invalidate Tegmark's CUH (Computable
    Universe Hypothesis) in that the universe, by even a narrow definition
    of universe, must contain consciousness, and, I presume, non-
    computability of consciousness would imply the CUH is false. That is,
    unless consciousness can have non-computable aspects that when
    "glued" (ultraproduct or some other method of "gluing"???) together
    throughout the universe, somehow (I know this is vague) the non-
    computable aspects of various parts of the universe all balance out to
    a computable universe. Hmm...things to think about... Maybe the CUH
    is true and brains work like quantum computers, somehow...?

    Anyway, Tegmark would be lending credence to his point by invalidating
    the proof of non-computability of consciousness for that relies on the
    "presumption" that consciousness is inherently a quantum process;
    obviously if their critical "presumption" is wrong, then their
    conclusion (consciousness not being computable) isn't necessarily so.

    I think it is worth splitting hairs here about the difference between
    consciousness and brain function but as of yet am aware of very little
    of the +formal+ theory behind either of these notions,
    philosophically, psychologically, or cognitive-scientifically.
     
    Last edited by a moderator: May 3, 2017
  23. Mar 6, 2008 #22
    What a can of worms.

    It is clear that scientific explanations are formulated using principles that are more powerful (in the sense of information or computational complexity) than the explanations themselves. This, in spite of the evident fact that these principles are informal and often, even usually, make mistakes.

    Accordingly, no scientific explanation can give a formally complete account of the causes of scientific explanations.

    This is what ultimately is bugging people like Penrose and Lucas. Let me explain or unpack this if I can.

    Kurt Godel formulated a version of what I am saying in a dilemma: either there are solvable Diophantine equations that nobody, no matter how smart or infinitely long-lived, could ever solve -- or we are not Turing-limited. People may quibble with Godel's assumptions, but they can't quibble with the actual dilemma.

    If there is something to what I am saying (and obviously I think there is), there are alternative situations we might be in. Nature could be super-Turing (i.e., might have computational complexity omega); or, we might not be completely natural.

    I see do not know of any empirical or practical way to distinguish between these alternatives. Quantum mechanics is relevant -- not by providing some non-computable functionality to the brain, but rather through non-locality!

    Naturally, there is no "formal" theory behind what I am saying -- by the very nature of the problem.

    But a number of people have grappled/are grappling with the issue. There are formal negative results, e.g. David A. Wolpert, Physical Limits of Inference [arXiv:0708.1362v1]. Especially the theorems concerning "self-aware devices." As far as I can tell (I am not an expert in this field) Wolpert's results amount to a refined formulation of Godel's dilemma. The Conway-Kochen Free Will Theorem [http://arxiv.org/abs/quant-ph/0610147] [Broken] also is relevant, since it depends on the assumption that we are free to choose any experimental setup from a set of possible setups -- but then non-locality infuses all of Nature with the super-Turing complexity (or, 'indeterminacy') of our choice.

    Again, if we're actually reasoning when we do science, then our reasoning powers are more powerful than any or all of their formal products. If we're not actually reasoning when we do science, what the hell are we doing?

    Finally, last thing I heard, you can't reason if you're not conscious.

    Regards,
    Mike Gogins
     
    Last edited by a moderator: May 3, 2017
  24. Mar 6, 2008 #23
    I can see how "self-aware devices" is definitely related to Tegmark's theories and the MUH however I am failing to see the connection between the rest of your post and the MUH.

    Thanks for pointing to the Physical Limits of Inference paper. I will definitely try to read this today, if possible.
     
  25. Mar 7, 2008 #24
    ...okay, so you have to be careful here. There is a lot of snake oil floating around concerning computability and consciousness, and some of it has even been published... :) This paper in particular is highly flawed. I will attempt to explain why I say this.

    The most basic problem here can be seen just in the introduction, where they lay out their strategy for the paper:

    Even before considering the rest of the paper, I can say that this approach is doomed to failure. This is why: Let us say you create a computational model of some physical phenomenon, say "consciousness" or "a hydrogen atom". You then prove this model to be computable or noncomputable. Have you just proven the computability of the physical phenomenon? Well, no. All you have done is proven the computability of your model. Whether anything has been proven about the physical phenomenon thus depends on whether your model accurately encapsulated the physical phenomenon.

    Under normal circumstances, my criticism here would be vacuous. With, say, a hydrogen atom, we know a LOT about its behavior and can thus build highly accurate models. We have a specific idea of what a hydrogen atom is. We can trap one in a lab. We can experiment on it. We have so much confidence in our understanding of hydrogen atoms that in a scientific work, adding "...assuming the standard theoretical model of the hydrogen atom is correct" is redundant.

    But consciousness is different-- because we don't know what a consciousness is. We don't have working theoretical models of consciousness. We can't trap consciousness and experiment on it, like we can with a hydrogen atom. We don't even have a clearly agreed-upon consensus of what consciousness means, or whether it exists or is just an illusion. Today most "models" of consciousness are provided by philosophers and religions and are not scientific in form.

    Neuroscience may someday be able to say something about consciousness in a scientific, rigorous way-- maybe in a hundred years, neuroscience will understand consciousness as well as we understand a hydrogen atom today. But we're not there yet. So until then, even if you say "I define X model of consciousness" and then prove things about X model, this does not count for anything unless you can also prove that X model describes something that exists in the universe and has something to do with the consciousness associated with humans. Doing this would be very hard, but if you don't do this then X model is just a straw man. Every attempt I've to date seen to prove the computability of "consciousness" at some point subtly incorporates these straw men.

    One more thing before considering the paper's actual arguments, several times the author of the paper attempts to give themselves an out by noting they believe they have not proven consciousness uncomputable, but merely uncomputable by a quantum computer. For example in the abstract: "The concept that this is solely a quantum effect and does not work for a classical case is also discussed." This is nonsense. If something can be computed by a quantum computer, it can be computed by a classical computer, and vice versa. The difference between a classical and quantum computer lies not in the class of problems that the device can solve, but in the problems that the device can solve efficiently. This difference is hugely important in practical terms, but it is not relevant to a computability argument. (And even this difference is technically only hypothesized, it is widely believed but BQP > P has not yet been proven.) This point however (i.e., does their purported proof also apply to the classical case?) is not relevant to the actual argument so let's set it aside.

    So, what about the actual arguments of the paper?

    --- --- --- ---

    The paper's arguments actually start in section 4, after an adequate introduction to the basic notion of computability. Section 4 says: "In order to represent a phenomenon of consciousness as a computational model, the manner in which a conscious activity is involved in a quantum system is first discussed." But look at how they do this. After introducing the standard definitions of qubits and observables, they say:

    What? So first off, this argument kind of seems to be dragging the Copenhagen interpretation in through the back door-- if you don't accept the dichotomy between a quantum system and a classical observer, the argument evaporates. This goes beyond just the Copenhagen interpretation here though-- the author here outright asserts that consciousness causes waveform collapse and that observables do not exist without a conscious observer. This is not the scientific consensus in quantum physics. It is the plot of a science fiction novel. It also seems kind of hard to see how this could even work-- if observables don't exist without a conscious observer, then what happened before conscious life evolved? Did quantum physics just proceed without any observables for the first 10 billion years?

    But most importantly think carefully about what they're doing here. They define consciousness as something which exists outside of quantum physics, independent of quantum physics. They begin by positing a notion of quantum physics where consciousness is above quantum physics in a sense, governing it. And now they are going to use this starting point to prove that a quantum system, a quantum computer, is insufficient to simulate consciousness-- in other words their goal is to prove that consciousness exists independent of quantum physics, outside of it. Even before presenting their argument, they are assuming their own conclusion, by embedding it into the interpretation of quantum physics they choose to work with! Even if you accept the consciousness-causes-collapse principle as literally accurate ("what the bleep do we know?"), this is not valid argumentation-- the reasoning is circular.

    Moving on, in section 5, "a quantum computational model is to be constructed such that it represents a phenomenon involving a conscious activity"; this is done so that the computability of that model can be analyzed. They first provide a definition for a very simple "quantum turing machine" with a 1-cell tape, apparently due to David Deutsch. The state of this QTM, they say, can be summarized by the ^µs , ^vs , ^µh, and ^vh, describing respectively the qubit for the input/output cell, the observable for the input/output cell, the qubit describing whether the machine has "halted", and the observable for whether the machine has halted.

    They then propose they will describe a "phenomenon" they call P2, in which "an observer observes" the input/output qubit rotates about the y-axis of its bloch sphere. Okay, fine. Then things get weird again:

    Why? I mean, why must the status of the observer have anything to do with consciousness, or the activity of the observer be necessarily conscious? The observer here is just a part of the formalism; from the perspective of the QTM, everything interesting about the observer is exhausted by the specified observable. Who or what the observer is is external to, and irrelevant to, the behavior of the QTM, which is just a mathematical object. As before, the paper argues that because results can be different for different observers based on their frame of reference in the Hilbert space, that those observers must be conscious. But this simply doesn't follow, and as I understand things the reference frame of the observer is anyway entirely specified by the observable, which is something that in this case the observer has no control over but is determined by the specification of the QTM. As far as the behavior of the QTM goes, the observer might as well not exist; the formalism of the QTM automaton is sufficient to encapsulate P2 and the observer adds nothing.

    Let's consider the end of their paragraph there, but simply delete the word "conscious" where it appears: "Therefore, similarly to the case with P1, the phenomenon, P2, necessarily involves an activity of the observer, represented as ^vs . Moreover, as discussed with the instance of P1, ^vs provides a full description of the status of the observer in reference to P2." This new text is just as valid! Consciousness is entirely unnecessary and irrelevant to the functioning and behavior of P2; the only reason why the observer has conscious status and conscious activity is because they defined their observer as being a conscious one.

    They then construct a quantum turing machine, T, which simulates the process P2, and assert:

    This sounds right, assuming we agree P2 doesn't necessarily contain or have anything to do with any sort of consciousness.

    Their argument, such as it is, completes in part 6. Here they define a final process, P3, which is just the same as P2, except instead of observing ^µs, the input/output qubit, they observe ^vs, the observable for the input/output qubit. ...Can you do that? I have no idea. Assuming though that observing the observable has some well-defined meaning in the QTM formalism, it's worth noting that this is nonsense:

    None of this means anything if you don't accept their initial premise, given in section 4 without justification or proof, that an observable implies a conscious observer. Even if we decide to consider only the case of a conscious observer, I don't think any of this follows; I don't see why (even if the observer is assumed to be conscious and even if it is possible in some meaningful way to measure an observable) we can say that the observable is being observed without measuring it. Experiencing the consequences of a reference frame in hilbert space is not the same as observing its value, it seems, and in any case the experiences of the observer are not part of the QTM formalism.

    Having defined P3, they assert that T, the QTM constructed from P2, is exactly the same as a QTM constructed from P3:

    Because I don't personally know what it means when you observe an observable, I will give them the benefit of the doubt and assume they are correct in their assertion that (they claim, due to the correspondence between the Schrodinger and Heisenberg pictures of QM) measuring the observable in P3 produces the exact same results in as measuring the qubit in P2.

    They then move in for the killing stroke:

    To me, this just sounds like they've proven that their claim of equivalence for P2 and P3 was mistaken-- they measured two different things evolving according to (bottom of page 8/top of page 9) two different evolution rules and the two different things had two different values. That doesn't sound very surprising. But as noted I'm giving them the benefit of the doubt on that point so let's just continue the paragraph, to the point they've been building to all along:

    Okay. Again, what? I don't know what, if anything, they proved by the contradiction they produced, but this contradiction is certainly not a consequence of anything to do with consciousness; even if the reason for the contradiction is the fact that ^vs is serving as both an observed quantity and an observable, they have done nothing to demonstrate that this observed/observable property is specially a property of consciousness; rather, it is simply a property of the problem they defined. Their proof ultimately applies just as much whether or not you choose to accept their premise that the observer in this particular scenario is conscious.

    So what on earth do we have here? We have a lot of handwaving asserting that observables imply a conscious observer. Then we shift gears and we have a proof that purports to first demonstrate that two processes P2 and P3 are identical, then demonstrate they are different. As I noted, there are some things I don't understand about their reasoning once P3 is introduced; I don't understand the meaning of this "observed observable", or whether that part of the proof even does mean anything, and if I've missed something here I'd be curious to know what. But I don't think my ignorance on this point matters, since it is not necessary to be able to criticize the proof itself to note that the proved point has no relationship to consciousness. They start out claiming they will provide a quantum model of consciousness and demonstrate it inconsistent. But even as dubious as this goal is, they never actually provide a model of consciousness, or a quantum system that simulates or necessarily incorporates consciousness. Instead they simply build a plain quantum model of computation and demonstrate that there was a flaw in the model they came up with.
     
    Last edited: Mar 7, 2008
  26. Mar 7, 2008 #25
    A discussion considerably less formal than this paper - bringing in mathematics and a bit of ontology but minus any mention of consciousness - is going on in [THREAD=215118]The Question : is mathematics discovered or invented?[/THREAD]
     
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