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In Shadows Of The Mind Roger Penrose puts forth a Gödelian argument against the computational theory of mind. He then goes on to suggest that quantum mechanics plays a central role in the realization of human consciousness. It is suggested that the argument about quantum processes in the brain falls short if we reject the original Gödelian argument, and there has been much literature that suggests we should. However, Penrose goes on to suggest that even if we deny the Gödelian argument we will still come to the same conclusion. His reason for this is that classical physics cannot effectively create conciousness (citing Libet).

If we do ignore the Gödelian argument and only pay attention to the latter, how does Penrose defeat the computational theory of mind with this argument? He states that quantum processes in the microtubules of neurons are the progenitors of consciousness, but how is this a non-computational process? The Church-Turing thesis tells us the limits of effective computation. The theory of quantum computation gives us explicit ways to represent quantum mechanical processes as computational processes. It was shown by Bernstein and Vazirani as well as Scott Aaronson (among many others) that quantum models of computation are able to be simulated by a Turing machine. With this being the case, how does Penrose show that the microtubules' quantum processes are non-computational?

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    How is "quantum processes in the microtubules of neurons are the progenitors of consciousness" any different from saying "chemistry in the microtubules of neurons is the progenitor of consciousness"? At the atomic shell level, quantum mechanics is completely understood and straightforward (if maybe somewhat "goofy"), and merely accounts for all chemistry. And neurons are so large, and neurotransmitter chemicals, etc, produced in such abundance, that reactions reach statistical equilibrium regardless of probabilistic interaction outcomes among individual molecules. – John Forkosh Dec 30 '16 at 23:34
  • That isn't the point of the argument. Yes, chemistry supervenes on physics so there isn't a difference in those two statements but that still doesn't explain why Penrose argues that it's not computational. If you're saying chemistry reaches statistical equilibrium that makes it even easier to give a computational model to describe it, which refutes Penrose's argument further. I don't understand what your objection is. – Not_Here Dec 31 '16 at 2:36
  • Okay, it's maybe I who don't understand the point to begin with. The whole computational analogy seems overblown. Anything whose behavior exhibits some kind of mathematical regularity can be used as a model of computation. Take checkers, place two on a scale, then place three more on the scale, and a measurement of their total weight models 2+3=5. Quantum computation's only claim to fame is that its particular mathematical behavior allows some computations to be performed with incredible speedup. But the underlying fact that it can be used as a model of computation is nothing special. – John Forkosh Dec 31 '16 at 5:01
  • @JohnForkosh I feel like you do misunderstand the argument because everything you are saying is exactly my point. Saying that anything that exhibits mathematical behavior can be a computational model is exactly what the computational theory of mind is based off of. It is the theory that the mind functions in a computational way. My question regards Penrose's argument against the computational theory of mind. He argues that quantum processes are not computational and as such the theory of mind is not computational because it relies on quantum mechanical properties. – Not_Here Dec 31 '16 at 5:19
  • I feel like you think that I am arguing for Penrose's thoughts. I disagree with Penrose's argument because I don't see how it shows that quantum mechanics is non computational. I asked this question to try and have someone explain what Penrose's argument is because I feel like I must just understand it. I feel as though you think I am arguing for his argument and you are trying to explain to me why his argument is wrong, which of course is what I already believe. I just wanted a clarification on what Penrose's thought processes is. – Not_Here Dec 31 '16 at 5:31
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The linked IEP article seems to me to be accurately summarized in the OP:"the argument about quantum processes in the brain falls short if we reject the original Gödelian argument... Penrose goes on to suggest that even if we deny the Gödelian argument we will still come to the same conclusion". But on my reading "the same conclusion" of Penrose is not that the computational theory of mind is false, but that quantum effects play a role in generating "consciousness". This is confirmed by Penrose's reference to Libet to the effect that classical physics is not up to the task. If we accept the Gödelian argument then there must be something beyond the classical mechanism on computational grounds, and quantum effects seem like the only alternative currently at hand. But there may be reasons independent of the Gödelian argument to assign them some role in the brain, and "consciousness" is one of them, it does not have to be linked to computational limitations or lack thereof.

If we reject the Gödelian argument, but still accept that classical physics is incapable of generating "consciousness", then quantum (gravity) effects can be taken to be the source of it by an inference to the best explanation (available). This does not, however, restore the Gödelian conclusion. Even if Penrose's "orchestrated objective reduction" in the microtubules has something to do with "consciousness" it is hard to see why those should necessarily give one means to outperform the Turing machine. It seems possible that conscious mind does not function computationally, but still can not outperform the Turing machine on computational tasks. Indeed, if the philosophical zombies are possible, as many proponents of irreducible intentionality and qualia believe, and they are bound by the Church-Turing thesis, then so will be any empirical performance by their conscious counterparts. So to keep both conclusions Penrose either needs to deny that even zombies are bound by the Church-Turing thesis, which would defeat the purpose since his whole point is that the Turing machine gets beaten through the intervention of "consciousness", or to join Dennett in denying the possibility of philosophical zombies.

Mind you, Penrose's speculations about quantum brain processes are based not on quantum mechanics, or even quantum field theory, but on his other speculations about quantum gravity, so he feels free to dismiss the computational limitations imposed by the existing work. In turn, his quantum gravity ideas are embedded into his Platonist metaphysics of "orchestrated objective reduction":

"An essential feature of Penrose's theory is that the choice of states when objective reduction occurs is selected neither randomly (as are choices following wave function collapse) nor algorithmically. Rather, states are selected by a "non-computable" influence embedded in the Planck scale of spacetime geometry. Penrose claimed that such information is Platonic, representing pure mathematical truth, aesthetic and ethical values at the Planck scale. This relates to Penrose's ideas concerning the three worlds: physical, mental, and the Platonic mathematical world. In his theory, the Platonic world corresponds to the geometry of fundamental spacetime that is claimed to support non-computational thinking."

The microtubules play a role similar to that of the pineal gland in the Cartesian mind-body dualism: the place where the ghost attaches to the machine. The idea that they seem to be one place in the brain where the quantum gravity might manifest was proposed by Hameroff and enthusiastically promoted by Penrose, the resulting fusion is called the Orch-OR model. This model makes a number of (in principle) testable predictions.

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    "Penrose claimed that such information is Platonic, representing pure mathematical truth, aesthetic and ethical values at the Planck scale." Wow! I knew Penrose was far out, but I didn't know that he was that far out. – Alexander S King Jan 6 '17 at 5:17
  • @AlexanderSKing It is more or less a cross between Heisenberg's "consciousness causes collapse" with Tegmark's universe made of math, inartfully expressed. But then he is not a professional philosopher. I give him credit for at least acknowledging that there needs to be an extra that animates the math. If Penrose were a philosopher he'd express himself more like Whitehead, with a quantum twist. – Conifold Jan 6 '17 at 5:45
  • On second thought, quantumness aside, how is what he says different from Chalmer's "consciousness must be fundamental" ? – Alexander S King Jan 6 '17 at 19:35
  • @AlexanderSKing Chalmers is a property dualist, Penrose is a thoroughgoing Platonist, not just about consciousness but about universals and even values. He also seems to be optimistic about prospects of physics of "consciousness" , while Chalmers contrasts conscious states to anything physical. – Conifold Jan 7 '17 at 22:20
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This is intended as a complement to Conifold's and Jobermark' answers

Penrose's argument can be broken down to two parts:

  1. Based on Lucas's Gödelian argument against mechanism, he argues that the human mind is more than just a Turing machine.
  2. The part of the human mind that is more than Turing machines can be explained by quantum phenomena in the brain.

If we do ignore the Gödelian argument and only pay attention to the latter, how does Penrose defeat the computational theory of mind with this argument?

So strictly speaking, the part of his argument that "defeats" the computational theory of mind has nothing to do with quantum phenomena. Nor is it really a new argument, he's only reviving an argument made by Lucas in 1959 (which can be simplified as: humans can see the truth of Gödel sentences but machines can't).

"Quantumness" comes into play to fill the gap between human minds and Turing machines, only after the gap has been already been established using Lucas' reasoning.

He states that quantum processes in the microtubules of neurons are the progenitors of consciousness, but how is this a non-computational process? The Church-Turing thesis tells us the limits of effective computation.

Here you have stepped into an ongoing controversy regarding the interpretation of the Church-Turing thesis: Is the Church-Turing thesis a broad result that puts a limit on any discrete and finite algorithmic process (a)? Or is it a very strict result about a specific class of models covered by Universal Turing Machines and Church's Lambda calculus - and its totally plausible that other models of computation might "break the Church-Turing barrier" or what is now called "Hypercomputation" (b)?

The majority opinion among mainstream computer scientists (i.e. those who specialize in computational complexity, the theory of algorithms, etc...) seems to be (a). In his thesis, Turing himself wrote that "It was stated ... that 'a function is effectively calculable if its values can be found by some purely mechanical process.' We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability† with effective calculability.". As you pointed out, Bernstein and Vazirani have shown that even quantum computers can't go beyond the Church-Turing limit and start solving undecidable problems.

To the theoretical CS crowd, talk of super-Turing computation is the their field's equivalent of perpetual motion machines and cold fusion. See Aaronson's lecture notes, as well as this article by Martin Davis - who was Church's student, and this article, by Andrew Hodges, a British mathematician who wrote a book on Turing.

On the other hand you will find that many outside of mainstream theoretical CS, subscribe to (b). As Aaronson himself says, google "Hypercomputation", and you will get dozens of articles and even entire conferences on the topic. In my own field of AI and Machine Learning, you frequently heard claims of some novel fuzzy logic or neural network approach which could potentially solve problems a Universal Turing Machine couldn't. Notably, Hava Siegelmann published in Science in 1995 that neural network based analogue computation could decide Turing-undecidable languages. Her results were disputed by Peter Shor, among others, and her result doesn't seem to have gained any traction beyond her own research.

In philosophy, it gets worse, with many philosophers arguing that Church-Turing is just a thesis and that non-Turing models of computation are entirely plausible, see for example here. In particular, Jack Copeland, a British philosopher who is also supposedly a Turing specialist, wrote an entry in the SEP called misunderstandings of the thesis, which purportedly refutes (a) (which Copeland calls 'Thesis M').

It doesn't help the confusion that another SEP article on Turing, written by the aforementioned Andrew Hodges, states the opposite (i.e. (a) is true).

Personally, as someone who started out in grad school working with Neural Networks and Fuzzy Logic, I bought into the various possibilities of hypercomputation, if only because I was attracted to the notion that my chosen field might be where the next big breakthrough was going to happen. As I progressed in my Ph.D, my understanding of computability and complexity theory got deeper, and I started to see why people like Aaronson and Davis don't take the idea of Hypercomputation seriously, and I am now firmly in the (a) camp. See this post in the theoretical CS SE.

What does this mean for Penrose's argument?

If you subscribe to (b), then his argument is totally plausible, but he doesn't even have to invoke anything quantum in order to defeat CTM, there are all sorts of ways that a human mind might be more powerful than a Turing machine, we just haven't worked out the details yet. See for example Paul Churchland's analysis of Penrose's argument in Chapter 9 of "The Engine of Reason, the Seat of the Soul", and his explanation of why quantum effects don't really matter to the argument. It should be noted that Churchland's argument as to why neural nets are capable of non-Turing computing is sketchy at best, and I don't agree with it. I mention it only as an illustration of possible response to Penrose.

If you subscribe to (a), then his argument is significantly more far fetched: At the very least, his speculations on non-Turing computable quantum processes, if true, would have serious consequences for physics and even math, since he would be effectively refuting Vazirani et al.

More likely, he wants to go further, and is implying a type of non-materialist dualism, a quantum gravity based argument that leads to a modern Platonism.

  • +1: you might be interested in oracle machines which are non-algorithmic machines which it is assumed that can solve problems in a certain complexity class in a single step - even undecidable ones; this is quite a common method, I've found in mathematics where limitations are gotten over; for example, adding irrationals to the rationals; or points at infinity. – Mozibur Ullah Jan 23 '17 at 23:13
  • @MoziburUllah Oracle machines are one option. Analogue computers are another, but neither of them are physically realizable. – Alexander S King Jan 23 '17 at 23:27
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Penrose believes that quantum mechanics is incomplete. So even if it is true that all quantum processes as they are currently known are computational, Penrose would argue there's something missing and that missing part is non-computational.

Reference video: https://www.youtube.com/watch?v=3WXTX0IUaOg

At 2:44, he explains his argument about why he thinks there's a gap in our understanding of QM, and that's where he guesses QM is outside computation.

  • Can you provide sources for this? – virmaior Dec 31 '16 at 12:33
  • I've added a video reference with Penrose talking. – Ameet Sharma Dec 31 '16 at 20:03
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Do you mean simulated or approximated? Unless you accept some simplifying theory like the holographic principle, the two are not the same.

If space does not come with a minimum resolution, or a maximum local entropy, no computation that inherently involves real numbers can be simulated on a digital computer without infinite storage capacity or unlimited speed. At that point the machine is no longer a Turing Machine.

In that case, chaos theory implies that chemistry cannot be fully computationally modeled, since molecules motion distributes energy continuously, in real, continuous space, even if their states are quantized. The result can only be approximated, and below the resolution of the approximation, tiny imbalances can make the whole approximation entirely wrong.

This obviously implies the same goes for the underlying solutions of wave equation, which is continuous across space, and classical quantum theory cannot be computationalized.

So if string theory is wrong about the ultimate granularity of usable space, and the equivalent of the holographic principle does not sneak back in for some other reason, Penrose's argument still has a point.

(I do not accept it, as it is an obvious instance of the 'God of the Gaps' delusion that two things that can't be explained and happen to be true must somehow be related. Just knowing physics is indeterminate would not imply humans have free will. It would only rule out the most simplistic arguments against it. I think the whole notion that physics and free will are related is an unhealthy obsession without a real meaning, as emphasized here.)

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