Discussion and review of Shadows of the Mind by Roger Penrose

Humankind has a fantastically deep and complex understanding of many of the scientific and philosophical puzzles that have baffled our ancestors. We understand the inner workings of the sun, and what lives in the most remote places on Earth; we have a good idea of what happened a fraction of a second after the Big Bang, and how distant galaxies were formed; and we can explain the basic structure and the evolutionary emergence of the complex organisms that roam our planet. But there are some mysteries that still elude us, and perhaps none are as close, personal, and important as the question of how our brains work – how they produce such a rich variety and complexity of thoughts and feelings, and how they perform such a huge range of tasks, from communication to mathematics to movement. And while AI is making great progress, our computers are still far behind most everyday cognitive tasks, such as the activity in your brain right now while you are reading this post. Furthermore, how can such a lump of meat produce the rich first-person experience that we are all consumed by throughout our lives?

Roger Penrose’s book Shadows of the Mind confronts these problems head-on, by giving a detailed assessment of the question: Is the human mind a computer? Can it be simulated by a computer, and can its workings be understood and explained only using computation? Penrose is conclusion is a resounding no: he concludes that the human mind is not a computer, and the workings of the human mind cannot be explained by computation alone.

This book is particularly interesting to me because, having read many books and listened to many talks/podcasts on consciousness and the mind, I myself have my own strong views on the answers to these questions. My conclusion is opposite to Penrose’s, namely, it is my belief that the human mind is performing a computation. Normally a conclusion such as Penrose’s would not bother me, because many arguments can be wielded that counter such claims. But Penrose’s book is different, because at its core is a solid logical argument, one that cannot just be ignored or countered easily and quickly.

In this post I will introduce Penrose’s logical argument, and then explain how he exploits the current gaps in the unification of general relativity and quantum mechanics to justify that the human mind must utilise a new type of physics – one that cannot be simulated on a computer.

The logical argument

I will dive straight in to the key argument at the centre of this book. Consider a computation that acts on a single natural number, n. This could be something as simple as the computation “n+1”. Or it could be a vastly complicated and elaborate computation, such as the algorithm that generates the graphics of the nth level on Super Mario.

Next, we need a way of labelling all the different computations that could act on n. We say that C(q,n) is the qth computation on natural number n. It is possible to write down every single possible computation that acts on a single natural number in this way. In other words, the complete set of computations labelled C(q,n) represents every single possible computation acting on a single natural number.

The way a computation works is that it takes in an input, then it runs for some period of time, finally producing an output. When it produces the output we say it stops. However, not all algorithms stop. For example, we could have an algorithm that takes n, then adds 1, then loops around and repeats this indefinitely – i.e. it keeps adding 1 forever. We say that this algorithm does not stop. Given this, it might be of interest to know whether a given algorithm C(q,n) stops or not. To do this, we can design another algorithm, A(q,n), whose job it is to say whether the qth computation on natural number n stops:

If A(q,n) stops, then C(q,n) does not stop.

This statement is true for any values of q and n. So what about taking q to be equal to n? With this, we get:

If A(n,n) stops, then C(n,n) does not stop.                                    (1)

But notice that considering only the cases when q is equal to n, the computation A(n,n) is now only acting on a single natural number, n. Because C(q,n) includes every possible computation and a single natural number, it must also include this computation. We can label it C(k,n). Therefore,

A(n,n) = C(k,n).

Now, this is true for any value of n. In particular, it is true for the case when n equals k. Considering this value only, we have:

A(k,k) = C(k,k).                                                                                                            (2)

Now consider equation (1) above, but for the specific value k. This reads:

If A(k,k) stops, then C(k,k) does not stop.                                    (3)

Finally, combining equations (2) and (3), we have:

If C(k,k) stops, then C(k,k) does not stop.                                    (4)

What can we conclude from this? Well, C(k,k) cannot stop, because if it did then according to equation (4) C(k,k) would not stop, which would be a contradiction. But A(k,k) (which is the same as C(k,k)) is the computation that is specifically designed to determine whether C(k,k) stops. Therefore, because A(k,k) cannot stop, it is not possible for a computational procedure to determine whether C(k,k) stops. But we know that C(k,k) does not stop. Therefore, we know something that no computational procedure could ever know!

Now comes the really interesting part: we can repeat the argument above, but instead of C(q,n) representing all possible computations, we can instead think of C(q,n) as representing all possible computational procedures available to humankind. We then find:

                  If C(k,k) stops, then C(k,k) does not stop.

                  Therefore we know that C(k,k) does not stop.

But C(k,k) cannot determine this, and therefore the collection C(q,n), which represents all possible computational procedures available to humankind, cannot determine that C(k,k) does not stop. But, as said above, we know that C(k,k) does not stop. Therefore, humans cannot be using a computational procedure to determine this, otherwise there will be a contradiction.

I have skipped some of the more subtle steps in the above explanation. The more precise conclusion would say: humans cannot be using a knowably sound algorithm to determine that C(k,k) does not stop (for brevity I will not introduce what knowably or sound mean – the interested reader can check out Penrose’s book!). To very briefly mentioned some of the more subtle details, instead of concluding that humans cannot be using a computational procedure, we could instead conclude that humans are not using a sound algorithm, or a knowably sound algorithm. But Penrose very convincingly argues that we are using a sound algorithm, and it is knowable, which allows to conclude that humans cannot be using a computational procedure to ascertain mathematical truth.

But if our brains aren’t working computationally, then what are they doing? Are we using some obscure algorithm that computer scientists haven’t come up with yet, or are our minds acting as a quantum computer? In fact, neither of these options are possible, because what we mean by computation includes any process in known physics, including quantum mechanics. Therefore, the inner workings of the human mind must be beyond current physics! So what could this physics be? Is there really any room for new physics, and even if there is, can it really solve the problem raised above?

Gravity-induced collapse of the quantum wavefunction

Those readers not already familiar with the measurement problem in quantum mechanics should read my earlier blog, as otherwise this section won’t make any sense! In short, to make any sense of the process of collapse of the wavefunction – and therefore to make any sense of quantum mechanics – one must subscribe to one of the many different interpretations of quantum mechanics. To choose between them, you must decide whether you think quantum mechanics is complete/correct or not, and whether you think the quantum wavefunction represents reality, or just our state of knowledge. My own view is that quantum physics is correct, and that the theory of decoherence explains why the quantum state appears to collapse. Then, taking the uncontroversial assumption that the quantum wavefunction represents reality, I (and many others) are led to the conclusion that there are multiple parallel universes!

But different people, with different backgrounds, expertise, prejudices, and life philosophies, can take a different perspective. Physicists have not yet successfully combined general relativity and quantum mechanics into a single unified theory, so in a sense they can’t both, in their current forms, be completely correct. Given this issue, a general relativity researcher’s view could be that general relativity is correct. As with quantum mechanics, general relativity has so far never been proved wrong. But as general relativity and quantum mechanics cannot be unified, one of them, or both of them, must be incorrect and in need of modification. And given the unintuitive conclusion of assuming that quantum mechanics was correct (parallel universes!), we should assume that quantum mechanics needs modification. Many scientists might argue that general relativity is more elegant and beautiful than quantum mechanics, so many researchers would think this is a reasonable stance.

How should we modify quantum mechanics? One of the issues of combining quantum mechanics and general relativity is that it is not clear how superpositions of different gravitational fields can be understood. Roger Penrose argues that this is solved by modifying the theory so that superpositions of gravitational fields are unstable – the larger the field, the more unstable – and therefore “large” objects like humans and cats cannot exist in superposition states.

Penrose then argues that there is reason to believe that a theory in which gravity collapses the wavefunction might be non-computational. Therefore, if the human mind utilises such physics, it will be working in a non-computational way.

This line of reasoning then solves many problems in one swoop: the measurement problem is solved because gravity causes collapse of the wavefunction; one of the main problems with unifying quantum mechanics and general relativity is solved because large gravitational fields can no longer be in a superposition; and the problem of what the human mind is doing if it is not using known physics is solved because the human mind might be using the new physics that comes from gravity-induced collapse.

New physics in the brain

A popular view nowadays is that neurons are the basic “computational” element in the brain. A single neuron is a type of cell that either fires (releases an electrical signal) or does not fire, depending on the electrical signals going into the neuron from other neurons connected to it at any given time. While a single neuron is quite limited, the theory is that the 100 billion neurons in the brain, linked together by 100 trillion connections, forms a highly complicated device that produces all the fantastic thoughts and intelligent processes that go on inside our heads. This view is backed up in part by the growing evidence that artificial neural networks – which try to replicate the brain structure – can do quite amazing things, such as recognise faces or play the game of Go better than the human champion. Furthermore, it has been showed that single neurons in the brain can react to specific individual’s faces (e.g. the Jennifer Anderson neuron! https://www.nature.com/news/2005/050620/full/news050620-7.html ).

But this view of how the mind works is bad news for Penrose’s view because neurons are too large and interact too much with their surroundings to sustain any quantum coherence for a decent amount of time. In other words, a neuron cannot both fire and not fire simultaneously, in a superposition, because it interacts so readily with its surroundings that this quantum superposition is almost instantaneously destroyed (https://arxiv.org/abs/quant-ph/9907009). And in order for gravity-induced collapse to play any significant part in the brains workings, a superposition state must be sustained for a reasonable amount of time. So if neurons are the basic element in the brain that are responsible for our thoughts, then gravity-induced collapse could play no part, and the brain would be entirely computational.

However, it has certainly not been proven that the entire working of the human mind is just a neural network – to some extent this is an assumption held by current science. Indeed, neurons are not the smallest things in the brain, and it is entirely possible that each neuron is itself a miniature processor, comprised of many more basic elements that together do the work (normally a good analogy here would be that each neuron is like a miniature computer, but as the whole point is that the brain is not acting as a computer, this analogy cannot be used!).

Things certainly gets quite speculative here – and Penrose himself admits this – but the proposed site for the required non-computational processes is something called a microtubule. I won’t go into what microtubules are, because I don’t understand myself, but what I do know is that they are small enough and sufficiently ill-understood that it is possible both that they play an important part in how our brain works, and that they can sustain quantum coherence long enough so that gravity-induced collapse can play a part in how they process information. Penrose’s theory then is that gravity-induced collapse takes place in the microtubules, which means that human brain is doing something non-computational, which in turn allows us to escape the grasp of the logical argument given earlier.

Penrose’s view has many significant implications for neuroscience and artificial intelligence. Firstly, we would not be able to understand how the mind works without understanding the new physics introduced above, and as we are very far from understanding this physics, a full understanding of the mind would be a very long way off. This seems to be quite opposing to the current view in neuroscience and consciousness studies. In addition, artificial intelligence that comes even close to human cognition would not be possible without understanding this new physics. Again, this is contrary to the current feeling in the artificial intelligence community, in which most researchers are confident that artificial intelligence will continue its impressive upward trajectory and before long many human-level cognitive tasks will be surpassed by artificially intelligent agents.

Can we escape Penrose’s logical argument?

Because of the stark contrast between Penrose’s view of how the mind works, and the artificial intelligence/neuroscience views, it seems important to try to resolve the issues raised in Penrose’s book. But while Penrose’s view is relatively comprehensive, well-structured, and well thought through, it depends heavily on his logical argument. It seems that if the logical argument falls down, then there is no strong scientific reason to believe that the human mind is acting non-computationally. In turn, there is no need to introduce something radical and speculative such as gravity-induced collapse in microtubules.

For many researchers, the whole theory seems to rest somewhat precariously on this logical argument, and it is highly tempting to just disregard it or ignore it. Unless I’m mistaken, this is potentially what almost the entire artificial intelligence and neuroscience communities are doing! To some extent this is understandable, because there aren’t many people in the world who understand logic and computation enough to thoroughly study Penrose’s argument in order to try and find a flaw in it. Until recently, I too was culpable of strongly believing that the human mind was a computer, without myself trying to find a flaw (or trying to find others who’d found a flaw!) in the logical argument. For such a large chunk of scientists to ignore his argument might turn out to be a risky strategy, and arguably more researchers should invest in studying his points more carefully.

I should mention here that after Penrose’s first book on this topic, The Emperor’s New Mind, in which he introduced a similar but slightly weaker logical argument, many researchers tried to find flaws, but Penrose has seemingly quite successfully batted these all away in Shadows of the Mind with a comprehensive half-chapter in which he carefully and thoroughly addresses 20 of such arguments. It’s probably safe to assume that Penrose also received many attempts at counter-arguments after his second book, but I know that Penrose himself has been unconvinced by these, and there is certainly no widely agreed upon and well understood counterargument that the rest of the scientific community wield in order to reject Penrose’s views.

Having now read the key parts of Shadows of the Mind twice, and spending some time pondering it, I myself now have many ideas for where I think the logical argument could fall down. However, I am not a logician and I have certainly not spent enough time to confidently say that I thoroughly understanding the argument, so while there is a remote chance that my ideas are correct, it’s probably more likely that they are not! So I’ll refrain from presenting them here, but I’m very happy to discuss them with anyone interested.

I will, however, present the core of a counter-argument introduced in a very interesting review of Shadows of the Mind by one of the most important consciousness researchers, David Chalmers (http://journalpsyche.org/files/0xaa25.pdf) . The general structure of Penrose’s argument involves taking a number of assumptions, then through the steps presented above he derives a contradiction. In Shadows of the Mind he quite convincingly argues that of all the assumptions are reasonable, except the assumption that the human mind acts computationally. Having derived a contradiction, at least one of the assumptions must be dropped, so Penrose drops the computational one. But Chalmers presents an argument (by McCullough and Löb) that takes all of Penrose’s assumptions, without the one about computation, then after a number of steps a contradiction is derived. Therefore, one of these assumptions is wrong. But this means that one of Penrose’s assumptions – and crucially not the one about computation – is incorrect, and Penrose should drop this one rather than his one about competition. Then there is no reason – at least not based on this logical argument – to think that the human mind is not acting as a computation.


I want to finish on a high, so I will end by saying that this is an exciting and fascinating book to read. It is clear, well-written, entertaining, and thorough. Given the conclusions alone, my first reaction was that they are implausible and outlandish. But following his arguments step-by-step, he only rarely takes radical or outlandish steps. If he is wrong, then he is only subtly wrong, so naturally the reader is led along a path that eventually leads to his conclusions. I still disagree with them, but I had an enlightening and thoroughly enjoyable time reading his book, and trying to work out which subtle points in the logical argument I disagree with, so that my own view – that the human mind is a computer – can be kept intact!


About P A Knott

I currently hold a Research Fellowship from the Royal Commission for the Exhibition of 1851. My research project will tackle a key challenge in the quantum technology revolution by designing computer algorithms that automate the engineering of useful quantum states. These algorithms will enable the design of novel experiments to bring forward the development of new technologies such as quantum computing, communications and metrology. In my previous post I worked at the University of Nottingham on a project entitled "Sentient observers in the quantum regime and the emergence of objective reality", with Gerardo Adesso, Marco Piani, and Tommaso Tufarelli. This project involved using quantum information theory to investigate foundational questions concerning the role of the observer in physical theories. More generally, my research interests include quantum metrology, quantum state engineering, quantum sensing networks, and optical interferometry.
This entry was posted in Artificial intelligence, Beyond quantum, Miscellanea, Quantum foundations, Science & Society. Bookmark the permalink.

2 Responses to Discussion and review of Shadows of the Mind by Roger Penrose

  1. Wojciech Kryszak says:

    Dear P.A. Knott,

    Thank you very much for this interesting review and concise presentation of Penrose’s reasoning.

    As for the computational mind and Penrose-Lucas ideas based on Goedel’s theorem,
    I have not seen better dismantling of (at least some of) such arguments than this one:


    All the best,
    Wojciech Kryszak


  2. P A Knott says:

    Dear Wojciech,

    Thanks for sending this, I’ll read it with interest!




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