Are you an algorithm? More specifically, is there an algorithm for consciousness?
Roger Penrose, Emeritus Rouse Ball Professor of Mathematics at Oxford University, and Stuart Hameroff, Director of The Center for Consciousness Studies at the University of Arizona, are approaching this question from two very different, yet strangely compatible angles.
Penrose’s two books, The Emperor’s New Mind and Shadows of the Mind, discuss the mathematical ins and outs of consciousness as a formal, logical system. This is an amazing and fascinating, albeit convoluted and technical, refutation of the assertions of AI specialists that a conscious machine is possible. Penrose leaves no options for AI by invoking the devastating theorem of Kurt Gödel, the Incompleteness Theorem. This powerful piece of mathematical artillery absolutely (and I mean, absolutely in the fullest sense of the term) destroys the possibility of there being a knowable, sound, consistent formal mathematical (or “logical”) system as the basis for conscious understanding. If there is no knowably true formal system, how is it that human consciousness has derived and understood many fundamental laws of science? Well, certainly not by means of any mathematical system, since all mathematical systems are shown to be self-contradictory by Gödel’s reductio ad absurdum. Therefore, consciousness is something which overflows the bounds of formal systems, including those from which any and all algorithms might be derived.
Penrose uses, as an example, Turing’s famous “stopping problem.” This problem asks if there is an algorithm which can determine whether any given computation does or does not stop, in that there seem to be some computations which do not stop but just carry on and on, never rendering an “answer.” According to Gödel’s theorem, there can be no algorithm to determine this, as any formal system which can be shown to soundly and consistently render an answer is demonstrated by reductio ad absurdum, to also not render an answer; in effect, the solution given by Gödel’s theorem is that if the computation stops then it does not stop. The analogous concept is rendered in plain language by the proposition, “This sentence is false.” If the sentence is true, it’s false and if it’s false, it’s true.
Penrose spends a great deal of time in both books on this argument. Somehow, says Sir Penrose, consciousness is able to become aware of mathematical truth even in the face of the powerful Gödelian assertion that truth cannot be formally derived (or “proven”). Even a child can discover that 1+1=2, but this awareness, this understanding of truth, is shown by Gödel’s theorem to be not computable, and therefore, it cannot be the result of an algorithmic computation. In Shadows of the Mind, Penrose adds nails to the coffin by raising every argument imaginable from AI and blowing them all completely out of the water. There is no algorithmic scenario that can overcome the awe inspiring purview of Gödel’s omnipotent theorem.
I know that I, for one, have never thought of it this way. Penrose notes that even Gödel himself did not believe that consciousness is fundamentally material, but that it is completely beyond and separate from materialism. Turing, though slightly less metaphysical, made the remarkable assertion that whatever algorithm might run consciousness, it would be an imperfect one, capable of making mistakes, learning, forgetting, etc. However, Penrose also has this covered, and even such “random” or “unknowable” algorithms are ruled out.
He then moves on to the altogether more interesting part of his thesis, involving the origin of consciousness in quantum processes. The upshot of his argument is that we do not have a mathematical system that is adequate to describe reality and this is the reason we do not have a science of consciousness. Schrodinger’s tatty equation is dragged out once again, and the huge, insurmountable problem of Objective Reduction becomes the focus of the second part of both books. Some fascinating mathematical digressions are made, especially with respect to the Second Law of Thermodynamics and Weyl’s theory, which places the problem of the original singularity, or “big bang” theory, in very sharp perspective. Simply reversing Schrodinger’s equation is not going to cut it for Sir Penrose.
So, both forwards and backwards, coming and going, Penrose takes the wave function to task. It is inadequate and cannot be considered a complete description of reality. What has this got to do with consciousness? In the coming days I will attempt to illuminate what I feel is the most important contribution to the scientific study of consciousness yet made in the combination of Penrose’s theory of Objective Reduction with Stuart Hameroff’s discoveries in brain science.
Stay tuned…same bat time, same bat channel.