Tegmark's Mathematical Universe

[Chronos]

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.

 

[marcus]

... the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. ...


I reflexively dismiss this idea ...

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.

 

[Chronos]

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.

 

[Coin]

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.

 

[Ben Niehoff]

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.

 

[Fra]

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

 

[staf9]

That seems like borderline philosophy to me

 

[Chronos]

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.

 

[Fra]

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

 

[marcus]

That seems like borderline philosophy to me

Borderline seems right. But could it be borderline something else?

 

[marcus]

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

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...

Chronos mentioned this in the original post. Untestability even in principle, the implication being that it doesn't 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.

 

[Fra]

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

 

[Fra]

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

/Fredrik

 

[Count Iblis]

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

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.

 

[setAI]

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" 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-

 

[Chronos]

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.

 

[Fra]

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.

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

 

[Chronos]

Just a side note - the paper was submitted to 'GR-QC' [of Arxiv], not 'physics.hist-ph'.

 

[PF_member_101]

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

 

[PF_member_101]

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.

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

- Hide quoted text -
- Show quoted text -
> A paradox?

> http://www.torrentreactor.net/torrents/1588942/Parallel-Worlds-Parall...

> There is one part when Tegmark is speaking, around the 27-30 minute
> mark or so, that they give a visual clue about parallel universes that
> was perhaps more interesting than the director realized, unless the
> director's assistant was Tegmark himself.

> When they showed two universes splitting, in one parallel, the
> Copenhagen interpretation is correct...and in the other, the Many
> Worlds interpretation is correct.

> There is a QM formula with [[[EXCEPT DURING OBSERVATION]]] in one half
> of the screen

> and

> in the other half of the screen, [[[EXCEPT DURING OBSERVATION]]] is +
> +crossed out++ by Tegmark.

> Interestingly, part of Tegmark's work says just that: not only do
> physical things split into parallels, but the laws of physics
> themselves are different in different universes.

> +++Therefore, The Copenhagen view is correct and the Many Worlds
> interpretation is correct.+++

> But which is correct in THIS universe?

> Or, maybe, that is a loaded question. More details on why that might
> be a loaded question has to do with my crew's speculation about there
> not just being parallel universes but also "overlaying" (or
> overlapping) of parallels, where the aggregate of parallels (aka, the
> universe) are (is) very much like the water system on earth: separate
> at times and other times, quite combined and overlaid upon one
> another. Indeed, if one "frog" is floating on the river, the "bird"
> sees the "frog" actually pass from the North Pole somehow through down
> to the Nile, passing thousands of different waterways in between, and
> the "frog" just thinks he has been in one body of water all along,
> which couldn't have been more wrong, at least, as far as the "bird"
> sees things.

> Then again, is there a bird's "bird?"

> And a bird's bird's bird?

> And a bird's bird's bird's bird?

> And do frogs have pets?

> Do those pets have pets?

> Do those pets have pets that have pets?

> Sound familiar? To me it sounds like a self-similar fractal and the
> way the universe would look if you started at a string and zoomed out
> to view the universe from the boundary of the universe, which might
> not "exist", unless the boundary of the universe exists
> mathematically, of course! I suppose one might want to push the
> envelope of mathematics to determine what the boundary of the universe
> is, to mightily abuse language.

> Well, assuming the MUH, this overlaying of parallels +must+ be the
> case due to the hierarchical nature of mathematics. Set theory is on
> a +somewhat+ lower echelon in the hierarchy than Category Theory,
> which is, on a lower echelon than Logic which is, in turn, on a lower
> echelon than Fuzzy Logic, a generalization of Logic. Perhaps instead
> of the ultimate set, I need to search for the ultimate math, but I
> think Logic and Model Theory and/or Cat might be that, except Logic
> does have its limitations, in some sense.

[This is now getting to the part relevant to the post I am replying to from PF]

The only problem is that Aristotle's mutual exclusivity might not
actually be universal, to resolve this apparent paradox. But even
within one parallel (mathematical structure?), ME (mutual exclusivity)
might be true in one region of space (ie, the context between and
containing mathematical structures), false in another, both true and
false in still another part of that parallel, and absolutely all
values of truth between true and false elsewhere in that parallel
universe. It seems somewhat mind boggling when pondering that.

In our "neck of the woods," I think ME is "almost" (sort of in a
Lesbegue measure sense) true. In other words, locally to myself and
probably you as well (whatever that might mean), the pseudo-well-
formed-formula below has a ++designated++ truth value in some truth
set D:
' for all wffs f, ( f & not(f) ||--> ^D) '

where ^D is the minimal element in D, or an arbitrarily chosen
representative of the ones of equally least value, respective of the
order on D. ^D is interpretable as the qualia FALSE.

In fewer words (in English):
"locally," D+( W(f)-->( f & not(f) ||--> ^D) )
where D+( ) means, "the truth degree of what follows is designated,"
and W( ) means, "what follows is a well formed formula," and ||-->
means there is a fuzzy logical sort of valuation function being
applied, and --> is the standard (in a fuzzy logical sense, of course,
but the truth set of this symbol definitely need not also be D--too
bad tex is not available to my knowledge here, that would make this
notation less unappealing to the eye) conditional connective. (I
think all of this is formalizable.)

I think in our dreams (double entendre intended), ME is "almost"
false, ie, D-( W(f)-->( f & not(f) ||--> ^D) ) where D-( ) means,
"what follows has an anti-designated truth degree."

Perhaps that could be related to the true difference between conscious
and unconscious.

Conscious could mean something like
X( D+( W(f)-->( f & not(f) ||--> ^D) ) )
and
unconscious could mean something like
X( D-( W(f)-->( f & not(f) ||--> ^D) ) )
where X( ) means something like, "in the context of the the parallel
network SAS labeled X is embedded or embeddable within, the following
is true."

 

[PF_member_101]

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" 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-

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.

 

[gogins]

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] 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

 

[PF_member_101]

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.

 

[Coin]

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.

...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:

In order to examine the computability of a physical phenomenon, the phenomenon should first be represented as a computational model; subsequently, the computability of this particular model can be examined. The physical phenomenon can then be claimed to be computable or not based on this examination. A similar approach will be taken in order to examine the computability of consciousness.

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:

Let us consider one particular phenomenon, denoted as P1, and described as follows: an observer observes the unitary evolution of a qubit, ˆμ, with respect to the observable, ˆν... When a measurement on ˆμ, with the observable, ˆν, is made, it yields a real eigenvalue that can be directly observed by the observer. Before discussing the description of the phenomenon, P1, using the dynamics of quantum theory, it is necessary to illustrate why the phenomenon, P1, necessarily involves a conscious activity of the observer... Because the measurement is performed by an observer, the observable is considered to be a coordinate or a reference frame of the observer, for a given qubit ˆμ. However, in quantum theory, observables, being a reference frame of the observer, are fundamentally different from reference frames in classical physics... Because the observables correspond to the reference frame of the observer and they exist in the complex Hilbert space, it must be concluded that, unlike reference frames in classical physics, quantum observables correspond to an observer's reference frame in thought. That is, an observable should be considered to represent the conscious status of an observer while observing a given state vector. This argument explains why the phenomenon, P1, necessarily involves a conscious activity... Because the evolution of observables through unitary transformations are performed in the Hilbert space and the observable is the observer's conscious status in P1, an observable that is being changed must correspond to a conscious activity of an observer.

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:

as in P1, the observer is observing the rotation indirectly and a measurement on ˆµs with the observable, ^vs , can be followed to confirm the evolution. Note that P2 is almost identical to the phenomenon, P1, except the unitary operation is specified as Uy . Therefore, similarly to the case with P1, the phenomenon, P2, necessarily involves a conscious activity of the observer, represented as ^vs . Moreover, as discussed with the instance of P1, ^vs provides a full description of the conscious status of the observer in reference to P2.

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:

Therefore, the phenomenon, P2, can be claimed to be computable because its computational representation, T , with the input ^µs , was shown to be computable by satisfying the criterion (A).

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:

Therefore, in P3 which describes consciousness of the observer, ^vs is serving the role of a state vector, because it is being observed, and an observable, because it is serving as the reference frame of the observer. Unlike the cases of P1 and P2, the measurement confirmation is not needed for P3. While the conscious status, ^vs , is evolving, the observer is not observing ^µs but ^vs . No measurement is needed in order to confirm the evolution of ^vs because the observer is already experiencing it as consciousness.

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 P3 is exactly the same as P2 except the input has changed to vector, ^vs , from ^µs , it follows that T , with an input, ^vs ,
must correspond to a computational model representing the phenomenon, 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:

Let us now discuss the computability of T (i) where i = T . In order for T (T ) to be computable... the halt qubit of T must have halted accompanied by a valid computation, i.e., both approaches should yield the same outcome
predicted in P3. However, the two approaches yielded two generally different outcomes for a single input vector, ^vs . The second approach did not yield the outcome described in the phenomenon, P3, because the vector is rotated by −δ. Therefore, this results in a contradiction because T halted on the invalid computation.

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:

The contradiction is noted to result from a peculiar property of consciousness in which ^vs is serving as a reference frame of the observer and as a system to be observed.

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.

 

[CaptainQuasar]

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

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]

⚛​

 

[PF_member_101]

...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.

abbreviated by me...

Very interesting and compelling arguments. Thank you for the time to lay out these arguments.

I didn't "want" consciousness to be non-computable, as that seems to falsify the CUH (Commutable Universe Hypothesis). I myself have been trying to falsify this paper you just seemed to falsify because my intuition tells me that the CUH is either correct or at least, in essence, true.

I don't imagine the CUH being completely false but what about atomic decay?

Isn't atomic decay activity noncomputable? Is it noncomputable or just, as of yet, we are unable to predict atomic decay behavior patterns? Are they seemingly chaotic or actually noncomputable?

This issue seems analogous [**] to proving a string of numbers is random in the http://en.wikipedia.org/wiki/Kolmogorov_complexity" sense, in that, if a sequence is not compressible, it is random, but it is in practice, impossible to know if a string of numbers is compressible unless one happens to compress it. If one spends 10 years trying to compress that seemingly random string (like numbers associated with atomic decay behavior patterns), and cannot, that does not prove that such processes are, in fact, non-computable.

I hope that the CUH can be proved to be fact. But I'm pretty sure no experiment can ever prove it. If the CUH is not falsifiable, then it is not science.

Is the CUH falsifiable?






[**] Another question is why is this a good or bad analogy? If the MUH (Mathematical Universe Hypothesis) is correct, then this is a good analogy because an atomic decay experiment is a mathematical structure (MS).

 

[gogins]

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.

Because Tegmark's theory requires Nature to be computable. The Godel dilemma, the physical limitations of inference, and the Free Will Theorem all pose a dilemma of the same general type: Either we are really thinking when we do science, but then Nature is incomputable; or else, we not really thinking when we do science, which is compatible with Nature being computable.

When I say "not really thinking", I mean that we may be conscious of thinking, but absolutely all of our thoughts are completely predetermined, such that our consciousness that we are exploring alternatives, deciding where to set up our measuring apparatus, trying different solutions to theoretical problems, and so on, is a kind of illusion.

"Really thinking," on the other hand, means that we really might decide to try harder, really might decide to give up on one particular hypothesis and try to develop another, really might decide to set up our apparatus on the south wall instead of the north wall.

I think it is very clear that this general type of dilemma is absolutely central to contemporary philosophy. Starting with Godel's dilemma and going on to the Free Will Theorem, philosophers and scientists have clarified the nature of the dilemma, to the point where it not only follows from metamathematics, it even follows from experimental observations of nonlocality and maximum signal velocity.

 

[Coin]

PF_member_101:

I didn't "want" consciousness to be non-computable, as that seems to falsify the CUH (Commutable Universe Hypothesis). I myself have been trying to falsify this paper you just seemed to falsify because my intuition tells me that the CUH is either correct or at least, in essence, true.

Well, personally I was picking at that point because as a CS person, the question of whether consciousness/reality is computable is of some interest to me... the CUH in specific, however, I'm less sold on :) I am actually myself still a little uncertain about the CUH and I think there is valid reason to question it. But I think some variant of the CUH is certainly true, I am just not sure yet about Max Tegmark's specific formulation of it.

I do think the MUH on the other hand basically has to be true-- in fact, I think you could argue the MUH is basically just being a more formal way of stating the naturalistic hypothesis...

I don't imagine the CUH being completely false but what about atomic decay?

Isn't atomic decay activity noncomputable? Is it noncomputable or just, as of yet, we are unable to predict atomic decay behavior patterns? Are they seemingly chaotic or actually noncomputable?

Okay, so this is kind of an important point you have raised! Physical laws do include randomness, and true randomness is considered "uncomputable". (In at least some important sense-- maybe this point can be contested depending on which definition of randomness we choose to adopt. As you note there can be algorithmic processes which mimic randomness quite convincingly in the sense of passing all the tests it occurs to us to attempt. At this point though I think we can probably invoke occam's razor-- a pseudorandom algorithm so nearly perfect it winds up passing all tests both human and implicit in physical circumstance, is arguably more of a stretch than just assuming there exists randomness.)

From a computability perspective, though, this isn't such a big deal. The reason why is that although proper randomness (IMO) can't come from a deterministic algorithm, we can simply expand our notion of computability to incorporate randomness as a primitive. In other words, we can just decide to use probabilistic turing machines instead of deterministic turing machines. This is a common thing to do in studying complexity/computability classes-- you imagine that your deterministic computer has access to an "entropy source", which it can consult at any time whenever it needs a bit of randomness. I actually have to admit that when I made my comment before about how quantum and classical TMs solve the exact same set of problems, I specifically had a probabilistic classical TM in mind-- at least just to compare apples to apples, since all quantum computers are probabilistic.


Now, interestingly, Tegmark's CUH uses deterministic, not probabilistic, turing machines, and he has his own way of getting around the question of whether randomness is computable. In fact the main problem I personally have with the CUH is it assumes a fairly simplistic model of computation! Anyway what Tegmark does is postulate that all randomness comes from some equivalent of the Many-Worlds hypothesis in quantum mechanics, such that "randomness" actually corresponds to a branching of parallel universes. From Tegmark's paper:

3. How the MUH banishes randomness

By insisting on a complete description of reality, the MUH banishes not only the classical notion of initial conditions, but also the classical notion of randomness. The traditional view of randomness (viewed either classically or as in the Copenhagen interpretation of quantum mechanics) is only meaningful in the context of an external time, so that one can start with one state and then have something random "happen", causing two or more probable outcomes. In contrast, the only intrinsic properties of a mathematical structure are its relations, timeless and unchanging. In a fundamental sense, the MUH thus implies Einstein's dictum "God does not play dice".

This means that if the MUH is correct, the only way that randomness and probabilities can appear in physics is via the presence of ensembles, as a way for observers to quantify their ignorance about which element(s) of the ensemble they are in. Specifically, all mathematical statements about probability can be recast as measure theory. For example, if an observer has used a symmetric quantum random number generator to produce a bit string written out as a real number like ".011011011101..." and if quantum mechanics is unitary so that the final state is a superposition of observers obtaining all outcomes, then in the limit of infinitely many bits, almost all observers will find their bit strings to appear perfectly random and conclude that the conventional probability rules hold...

Again I'm not sure I like this solution but from a philosophical perspective I think this is solid.

Is the CUH falsifiable?

I do not think so.

The question of how to contrive a physical model which is explicitly uncomputable is kind of an interesting one; I'm actually not sure right now if it can be done.

 

[Q_Goest]

I don't imagine the CUH being completely false but what about atomic decay?

Isn't atomic decay activity noncomputable? Is it noncomputable or just, as of yet, we are unable to predict atomic decay behavior patterns? Are they seemingly chaotic or actually noncomputable?

Okay, so this is kind of an important point you have raised! Physical laws do include randomness, and true randomness is considered "uncomputable". (In at least some important sense-- maybe this point can be contested depending on which definition of randomness we choose to adopt.

Note that Tegmark discusses this when he talks about the bird and the frog on pg 3, ending on pg 4:

Fundamental randomness from the bird's view is by definition banished.

He says this because given MWI, everything is deterministic 'from the bird's view'.

I don't see anywhere that he addresses MWI specifically in order to do away with randomness being uncomputable, but it certainly looks like that's what he wants.

 

[PF_member_101]

Note that Tegmark discusses this when he talks about the bird and the frog on pg 3, ending on pg 4:


He says this because given MWI, everything is deterministic 'from the bird's view'.

I don't see anywhere that he addresses MWI specifically in order to do away with randomness being uncomputable, but it certainly looks like that's what he wants.

By Tegmark's theories, isn't the MWI true in some parallels and false in others?

 

[gogins]

Note that Tegmark discusses this when he talks about the bird and the frog on pg 3, ending on pg 4:


He says this because given MWI, everything is deterministic 'from the bird's view'.

I don't see anywhere that he addresses MWI specifically in order to do away with randomness being uncomputable, but it certainly looks like that's what he wants.

Randomness is indeed uncomputable. Note that nonlocality REQUIRES uncomputable randomness. In other words, uncomputable randomness is a central assumption of standard interpretations of quantum mechanics. It can be approached from several angles, but they all have to do with observation.

As far as I know, uncomputable randomness can be banished from Nature only by introducing hidden variables in its place.

The Free Will Theorem was derived just in order to clarify this.

Come on -- admit that Nature is uncomputable. It doesn't mean we can't be scientists any more. It just means that Nature is uncomputable. Uncomputable is not "irrational" in the sense of anti-mathematical or anti-scientific.

This whole discussion reminds me of the ancient Greeks being freaked out by the square root of 2 and other irrational numbers. Since Godel a lot of people are freaked out by uncomputability. OK, we have irrational numbers. We finally got used to that. OK, now we have uncomputable numbers.

Nature is numerical -- that much is clear, right? Why should Nature be numerical, and involve transcendental and irrational numbers, but shrink from uncomputable numbers? What reason is there for such an assumption?

Regards,
Mike Gogins

 

[Q_Goest]

Randomness is indeed uncomputable. Note that nonlocality REQUIRES uncomputable randomness. In other words, uncomputable randomness is a central assumption of standard interpretations of quantum mechanics. It can be approached from several angles, but they all have to do with observation.

As far as I know, uncomputable randomness can be banished from Nature only by introducing hidden variables in its place.

Hi Mike. That's not what Tegmark is saying (ie: you're looking at it from the "frog's" perspective). His reference to MWI being determinate overall is commonly accepted as he states. If you've read his paper, and the portion especially around page 3 and 4, what is it you don't understand?

 

[PF_member_101]

Someone in this thread mentioned an article that I think could lead to something connected to Tegmark's Self Aware Structures (SAS's). Since it may not be directly relevant to THIS particular discussion, I stared a new thread for anyone interested in the math section:

https://www.physicsforums.com/showthread.php?p=1640031#post1640031"

 

[gogins]

Hi Mike. That's not what Tegmark is saying (ie: you're looking at it from the "frog's" perspective). His reference to MWI being determinate overall is commonly accepted as he states. If you've read his paper, and the portion especially around page 3 and 4, what is it you don't understand?

Tegmark uses the word "random" much, much too loosely. He does not appear to address random quantum state transitions, nonlocality, or anything like the Free Will Theorem except by appeal to the MWI. He can do this because he assumes that any mathematical structure that could be a universe has finite computational complexity. So, Tegmark simply begs the question. He is pulling a bait and switch operation. The randomness at stake in uncomputability or in physics has to do with complexity at root, and nothing to do with the MWI. A statistically random temporal sequence MIGHT be random in the sense of computational complexity -- or it might be a pseudo-random sequence. But an incomputable sequence WILL be statistically random.

Actually Tegmark acknowledges the problem here:

"A convincing
demonstration that there is such a thing as true randomness
in the laws of physics (as opposed to mere ensembles
where epistemological uncertainty grows) would therefore
refute the MUH."

Now, let us take the ERH and the MUH seriously by including in it mathematical objects of transfinite complexity. Again, why not? Why stop at finite complexity? This is precisely what is begging the question. If mathematics has omega complexity (and we know it does), and if Nature is mathematics, then doesn't Nature have omega complexity if MUH is true?

Tegmark implicitly acknowledges the problem by distinguishing between the Level IV multiverse which is the union of uncountably many finite computable universes (and so Level IV is incomputable), and our universe (which is a computable universe), but what is it that physically acts to create this distinction? (third man argument). If it acts physically it invalidates the CUH, but if it does not act physically then there is no distinction between Level IV and our universe, which is thus incomputable.

Regards,
Mike Gogins

 

[PF_member_101]

I believe I have a working candidate for a plausibility case for a
structure being literally the universe, assuming the MUH.

It is the structure U(U), where the first U is script and the second
is blackboard bold, on page 3 of the following document, listed under
"conjecture 4."