I spent a year as a guest of Penrose’s biologist collaborator, Stuart Hameroff, at the University of Arizona, and my one peer-reviewed publication dates from that time, so I can tell you more than you want to know about this subject. :-)
First you should understand the order of events. Penrose published his book arguing that there should be a trans-Turing quantum-gravity process happening in the brain. Then Hameroff wrote to him and said, I bet it’s happening in the microtubules. Thus was born the version of the idea that most people hear about.
Penrose’s original argument combines an old interpretation of Gödel’s theorem with his own speculations about quantum gravity. The first part goes like this: For any mechanized form of mathematical reasoning, there are, necessarily, mathematical truths which it cannot prove. But we can know these propositions to be true. Therefore, human cognition must have capabilities which are not Turing-computable.
In the second part, Penrose observes that the whole of nongravitational physics is Turing-computable, but that gravitational physics is at least potentially not, because it may involve quantum sums over arbitrary 4-manifolds, and topological equivalence of 4-manifolds is not Turing-decidable. He also introduces one of his own physical ideas: Hawking evaporation of black holes appears to involve destruction of quantum information, so he proposes that conservation of probability flow is maintained by nondeterministic wavefunction collapse, which creates quantum information. He also has a technical argument against the possibility of superpositions of different geometries. So, if there are mesoscopic quantum superpositions in the brain whose components evolve towards mass distributions (and hence local space-time geometries) sufficiently different from each other that the superposition must break down, then, there is an opportunity for trans-Turing physical dynamics to play a role in human cognition.
The physical argument is very ingenious but probably wrong in two out of three places. But first, how about the prior argument using Gödel? There are two key considerations here.
Firstly, the true propositions which a formal system cannot itself prove can be proven, if you know the interpretation of the formalism, and if you know the axioms to be true and the methods of inference valid under that interpretation. In other words, knowing the semantics of the system is what allows you to construct the undecidable propositions and have an opinion about their truth. The logician Solomon Feferman has shown that if you have an extra logical primitive, “logical reflection”, which amounts to accessing this information about meanings, then there are no undecidable propositions. The combination of a valid formal system and indefinitely iterated logical reflection gets you everything.
Secondly, this makes it plain that there is a connection between the Penrose-Gödel argument, and John Searle’s problem regarding the semantics of computational states. If a thought is actually a brain state, what is it about that brain state that makes it a thought about one thing rather than another? Penrose doesn’t address this issue, yet Feferman’s analysis makes it clear that it’s metacognition or reflective cognition about meanings which produces Gödelian insights.
It is possible to attack Penrose’s ultimate conclusion by saying there’s no empirical evidence that humans can engage in logical reflection of arbitrary order. (The higher iterations of logical reflection correspond to transfinite ordinals, because they involve induction over infinite axiom sets.) If humans can only logically reflect up to order N, then a formal system of order N+1 should be capable of equaling the human ability to reason. But really, the conclusion I draw is that we will see no end to this particular dispute until we understand how neurocomputational semantics works. Until then, we simply can’t offer a neurocognitive account of advanced mathematical reasoning.
As for the physical arguments, I try to judge them from the perspective of string theory. The bit about sums over arbitrary 4-manifolds might be true; string theory is a work in progress, like most particle physics theories it’s known and used only in an approximate form, and this is a level of detail which presently is neither used nor understood. On the other hand, black hole evaporation is a unitary process in string theory, so the ingenious idea of wavefunction collapse balancing quantum information loss loses its motivation. As for the technical argument about geometric superpositions, that only applies if you think superpositions are objective physical states rather than generalized probability distributions. If you take the latter view, the argument loses its potency.
Now, microtubules. My grasp of molecular neuroscience is a whole lot less than my grasp of physics, but it’s definitely true that neuronal microtubules are not thought to play much of a role in cognition or consciousness. Microtubules are a dynamic structural organelle. They are scaffolding for the transport of vesicles, they move the chromosomes around during cell division, they are involved in pseudopod extrusion and cell motility. They occur in all your cells, not just neurons. Because a neuron is just another cell, but one which has specifically been shaped to perform an information-processing function, it’s not surprising that microtubules are involved in the execution of that function. But everything known suggests it’s a peripheral involvement.
I ended up in Arizona because I had my own reasons for being interested in quantum brain theories. And I’ll say this much in favor of microtubules: if you are looking for a molecular structure in the brain which might contain long-lived quantum states, the microtubule is a great candidate. It gives you a two-dimensional space (a cylinder) protected from environmental interaction by the tails of the tubulins. A lot of cool quantum things can happen in two dimensions. The problem is, how would it be relevant to anything cognitive?
Penrose and Hameroff wrote some papers applying Penrose’s quantum-gravity collapse model to microtubules. I don’t believe those calculations apply to reality. I’ve also mentioned why, even if you could show that quantum coherence does exist in the microtubule, that doesn’t yet connect it to conscious cognition. But I will still put in a word for Penrose’s original conception of quantum-gravitational dynamics maybe playing a part in the physics of cognition.
If one does wish to suppose—as I do—that the neural correlate of consciousness is actually a quantum state of some brain subsystem, rather than a coarse-grained classical computational state; if one does suppose that the manifest attributes of conscious experience are to be identified with fundamental degrees of freedom in that quantum object; then it is logical to suppose that some of those degrees of freedom are what we would call, from a physical perspective, gravitational, and that they might even be dynamically relevant. The idea that Feferman’s operation of conscious logical reflection is computationally implemented by a gravitational subalgebra of the full set of physical transformations of state… that’s my version of Penrose’s idea. I certainly don’t regard it as a logical necessity; it’s just a stimulating hypothesis. I look forward to the day when we know enough that I can actually rule it in or out.
Excellent explanation, thanks! So if I’m understanding correctly, while there are severe problems with Penrose’s theory, it’s not in the category of things to be casually dismissed as bunk; experts have found it an interesting line of thought to investigate, at least.
I spent a year as a guest of Penrose’s biologist collaborator, Stuart Hameroff, at the University of Arizona, and my one peer-reviewed publication dates from that time, so I can tell you more than you want to know about this subject. :-)
First you should understand the order of events. Penrose published his book arguing that there should be a trans-Turing quantum-gravity process happening in the brain. Then Hameroff wrote to him and said, I bet it’s happening in the microtubules. Thus was born the version of the idea that most people hear about.
Penrose’s original argument combines an old interpretation of Gödel’s theorem with his own speculations about quantum gravity. The first part goes like this: For any mechanized form of mathematical reasoning, there are, necessarily, mathematical truths which it cannot prove. But we can know these propositions to be true. Therefore, human cognition must have capabilities which are not Turing-computable.
In the second part, Penrose observes that the whole of nongravitational physics is Turing-computable, but that gravitational physics is at least potentially not, because it may involve quantum sums over arbitrary 4-manifolds, and topological equivalence of 4-manifolds is not Turing-decidable. He also introduces one of his own physical ideas: Hawking evaporation of black holes appears to involve destruction of quantum information, so he proposes that conservation of probability flow is maintained by nondeterministic wavefunction collapse, which creates quantum information. He also has a technical argument against the possibility of superpositions of different geometries. So, if there are mesoscopic quantum superpositions in the brain whose components evolve towards mass distributions (and hence local space-time geometries) sufficiently different from each other that the superposition must break down, then, there is an opportunity for trans-Turing physical dynamics to play a role in human cognition.
The physical argument is very ingenious but probably wrong in two out of three places. But first, how about the prior argument using Gödel? There are two key considerations here.
Firstly, the true propositions which a formal system cannot itself prove can be proven, if you know the interpretation of the formalism, and if you know the axioms to be true and the methods of inference valid under that interpretation. In other words, knowing the semantics of the system is what allows you to construct the undecidable propositions and have an opinion about their truth. The logician Solomon Feferman has shown that if you have an extra logical primitive, “logical reflection”, which amounts to accessing this information about meanings, then there are no undecidable propositions. The combination of a valid formal system and indefinitely iterated logical reflection gets you everything.
Secondly, this makes it plain that there is a connection between the Penrose-Gödel argument, and John Searle’s problem regarding the semantics of computational states. If a thought is actually a brain state, what is it about that brain state that makes it a thought about one thing rather than another? Penrose doesn’t address this issue, yet Feferman’s analysis makes it clear that it’s metacognition or reflective cognition about meanings which produces Gödelian insights.
It is possible to attack Penrose’s ultimate conclusion by saying there’s no empirical evidence that humans can engage in logical reflection of arbitrary order. (The higher iterations of logical reflection correspond to transfinite ordinals, because they involve induction over infinite axiom sets.) If humans can only logically reflect up to order N, then a formal system of order N+1 should be capable of equaling the human ability to reason. But really, the conclusion I draw is that we will see no end to this particular dispute until we understand how neurocomputational semantics works. Until then, we simply can’t offer a neurocognitive account of advanced mathematical reasoning.
As for the physical arguments, I try to judge them from the perspective of string theory. The bit about sums over arbitrary 4-manifolds might be true; string theory is a work in progress, like most particle physics theories it’s known and used only in an approximate form, and this is a level of detail which presently is neither used nor understood. On the other hand, black hole evaporation is a unitary process in string theory, so the ingenious idea of wavefunction collapse balancing quantum information loss loses its motivation. As for the technical argument about geometric superpositions, that only applies if you think superpositions are objective physical states rather than generalized probability distributions. If you take the latter view, the argument loses its potency.
Now, microtubules. My grasp of molecular neuroscience is a whole lot less than my grasp of physics, but it’s definitely true that neuronal microtubules are not thought to play much of a role in cognition or consciousness. Microtubules are a dynamic structural organelle. They are scaffolding for the transport of vesicles, they move the chromosomes around during cell division, they are involved in pseudopod extrusion and cell motility. They occur in all your cells, not just neurons. Because a neuron is just another cell, but one which has specifically been shaped to perform an information-processing function, it’s not surprising that microtubules are involved in the execution of that function. But everything known suggests it’s a peripheral involvement.
I ended up in Arizona because I had my own reasons for being interested in quantum brain theories. And I’ll say this much in favor of microtubules: if you are looking for a molecular structure in the brain which might contain long-lived quantum states, the microtubule is a great candidate. It gives you a two-dimensional space (a cylinder) protected from environmental interaction by the tails of the tubulins. A lot of cool quantum things can happen in two dimensions. The problem is, how would it be relevant to anything cognitive?
Penrose and Hameroff wrote some papers applying Penrose’s quantum-gravity collapse model to microtubules. I don’t believe those calculations apply to reality. I’ve also mentioned why, even if you could show that quantum coherence does exist in the microtubule, that doesn’t yet connect it to conscious cognition. But I will still put in a word for Penrose’s original conception of quantum-gravitational dynamics maybe playing a part in the physics of cognition.
If one does wish to suppose—as I do—that the neural correlate of consciousness is actually a quantum state of some brain subsystem, rather than a coarse-grained classical computational state; if one does suppose that the manifest attributes of conscious experience are to be identified with fundamental degrees of freedom in that quantum object; then it is logical to suppose that some of those degrees of freedom are what we would call, from a physical perspective, gravitational, and that they might even be dynamically relevant. The idea that Feferman’s operation of conscious logical reflection is computationally implemented by a gravitational subalgebra of the full set of physical transformations of state… that’s my version of Penrose’s idea. I certainly don’t regard it as a logical necessity; it’s just a stimulating hypothesis. I look forward to the day when we know enough that I can actually rule it in or out.
Excellent explanation, thanks! So if I’m understanding correctly, while there are severe problems with Penrose’s theory, it’s not in the category of things to be casually dismissed as bunk; experts have found it an interesting line of thought to investigate, at least.