Gödel, Penrose, and Consciousness

Brilliant physicist and Nobel Laureate Sir Roger Penrose argues, using Gödel’s Theorem, that consciousness cannot be computational. In other words, there is more to human consciousness than can currently be explained scientifically.

Gödel’s theorems are among the most important—and most difficult to understand—breakthroughs in modern mathematics, and science in general. They are related to the vexing problem of self-referring language, as discussed in the classic book

Gödel’s theorems are among the most important—and most difficult to understand—breakthroughs in modern mathematics, and science in general. They are related to the vexing problem of self-referring language, as discussed in the classic book Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.

As an example of the problem, take the following statement:

“This sentence is false.”

If the statement is true, then it’s false; and if its false, then it’s true. At first sight this might appear to be just a silly trick of self-referential language, but the problems associated with self-reference are not at all trivial. They formed the basis of Gödel’s deep investigation into the theory of arithmetic systems.

Gödel’s two “incompleteness theorems” say that any logical system contains either contradictory statements or statements that cannot be proven. A consistent logical system is composed of a set of axioms which allows you to do arithmetic according to rules related to the axiomatic statements. Gödel found that in any such system there must be at least one statement which is unprovable using just that system’s axioms.  

Gödel was able to prove these theorems, in a way that is far beyond my understanding. What Penrose says (for example, in this recent interview with Jordan Peterson) is that Gödel’s theorem shows that consciousness cannot be fully explained by any kind of numerical computational process. https://www.youtube.com/watch?v=Qi9ys2j1ncg

I do not grasp all the details of how this works, but here is something he said in the interview: “Understanding—whatever that word means—is not computational. It’s not the following of rules. It’s something else.”

The implications of this idea are quite staggering. For one thing, it suggests that while AI might become very intelligent, it could never become fully conscious, in the human sense. It also raises the question of how consciousness could arise in human (and perhaps other) brains if things like computational complexity cannot explain all of it. Indeed, it makes the task of even defining consciousness in a formal scientific sense difficult.

I might be overstating the implications and might not even have a correct understanding of what Sir Roger was saying. But these are my impressions. Corrections in comments are entirely welcome.

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12 Responses to Gödel, Penrose, and Consciousness

  1. Fascinating idea, Sy! Thanks for the link to Penrose’s interview so I can get the details.

    Here’s a trivial correction: Hofstadter’s book is Gödel, Escher, Bach: An Eternal Golden Braid. You wrote Escher, Gödel….

  2. Interesting, Sy. So, have you been reading my passing comments with “estebang”?

    As for Hofstadter’s position, it’s one of emergence, weak versus strong. Are we automatons merely deluded by the circularity of the process, or does this self-referencing complexity result in something emergently real? I think Hofstadter falls toward the former (which he clarified in, “I am a Strange Loop”), though it remains a vexing question. (For reasons of my own, I’ve come to suspect that it’s a false dichotomy.) However, I’ll confess that I’ve all but given up on free will, which I think physics treats as a Sacred Cow when what QM describes is actually a variation on the halting problem arising from von Neumann entropy. But the strong emergence of consciousness is something else. Hofstadter’s position is at least philosophically consistent; though even as a determinist, there’s still room for both strong emergence as well as an observer.

    Respectfully, I disagree with your conclusion regarding artificial intelligence”. It’s not a matter of whether or not such an intelligence is “conscious” (which as Hofstadter observes is a rather nebulous assertion anyway). Rather, it’s whether or not both we (Turing test), and perhaps more importantly, the intelligence itself *believe* that it’s conscious; that is, whether or not it’s “experiencing” its own existence in a sort of Turing self-test. “Structure is everything.” (-Julian Barbour) Even if consciousness is indeed a strongly emergent and observational characteristic, it still emerges from the information in patterns. Both Hoffstadter’s and Penrose’s assertions are merely that such information may result in something fundamentally not computable, and thus strongly emergent.

    • Kumi,

      I realized I have 2 previous blog posts dealing with emergence, but I never made the connection between emergence and non computability, so thanks for that. If you type emergence into the search box, you can find those earlier posts.

  3. dgilmanjm says:

    Having dabbled a bit in programming, I disagree with Lightness Travelling. For the intelligence itself to *believe* that it’s conscious would require the programmer to set what is to be interpreted as such, making it an illusion, not a reality.

    • You can call me “Kumi”. (You and Sy both knew me back at Gather.)

      • dgilmanjm says:

        Well, nice to become reacquainted.

        One thing I should add is that on another thread, I had expressed the belief that it is the very design of the brain that creates consciousness. That belief, in my opinion is part of that very same theory that I am actually struggling very hard to comprehend.

    • At Gather, I used the nom de guerre of a 416-year old witch from a Philip Pullman novel.
      I’ve wrestled with the same thing. Clearly, experience and consciousness somehow emerge from the structure of the brain. So I don’t have any problem with attributing “consciousness” to an inanimate dynamic system. The question of whether or not it’s an “illusory” phenomenon is, to me, a case of it’s being reduceable to a purely mathematical (computable) description. And from even the most brutally objective, deterministic version of nature, it’s not clear that’s even hypothetically possible.

      • dgilmanjm says:

        “And from even the most brutally objective, deterministic version of nature, it’s not clear that’s even hypothetically possible.”
        Whereas I do not view that nature is deterministic; also, my opinion is that consciousness cannot be reduced to a purely mathematical (computable) description. That is the belief that is unprovable. In my programming days, I was frustrated to discover that in computing, there is no true random. That I assume is the main reason for your conclusion that nature is deterministic.

    • “Pseudorandom” numbers… yes. “Complex Chaotic Systems” (Edward Norton Lorenz) encompassed my graduate work. Rather sucked the air out of the whole concept of an “independent observer” (as well as the whole quantum woo thing). From the physics perspective, the question is reductionist, even if only in principle.

      But with the caveat that we certainly don’t understand far more than we do, it’s actually not clear that all physical phenomena are in fact mathematically reduceable, at least under certain conditions. A technicality maybe little more than jargon here, but something associated with what are known as “coupling constants” is an example. If perhaps just an artifact of the maths, they at least raise the possibility that the complexities of atomic physics, chemistry and biology give rise to something strongly emergent, or, “…the hypothetical possibility that a system with many constituents displays novel behavior which cannot be derived from the properties and interactions of the constituents.” (-Sabine Hossenfelder) I could go on; but it’s not my intent to hijack Sy’s post. 😉

      • dgilmanjm says:

        Love this; especially because I agree with, “the possibility that the complexities of atomic physics, chemistry and biology give rise to something strongly emergent, or, ‘…the hypothetical possibility that a system with many constituents displays novel behavior which cannot be derived from the properties and interactions of the constituents.’”

        I actually tried to work in the uncertainty principles associated with the Quantum Theory, but my skills at math are not good enough for me to do that well 😀

  4. Arnold says:

    Consciousness of God is via Christ, “God Incarnate.” He died and arose and ascended so to give us his life: personal relationship is what he most desires. He wants to share himself and be known for no other reason than, himself. ‘Christ is come in the flesh.’
    Conscious of God how? John put it in 1John1.7, ‘If we walk in the light of his eyes we have fellowship with one another.’ Fellowship is in the common details of life- in the sufferings, conflicts, boredoms and joys. Just as Jesus walked with his Father.

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