Shadows of the Mind: A Search for the Missing Science of Consciousness

Shadows of the Mind: A Search for the Missing Science of Consciousness is a 1994 book by mathematical physicist Roger Penrose, and serves as a followup to his 1989 book The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics.

In the book, Penrose expounds upon his previous assertions that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine-type of digital computer. Penrose hypothesizes that quantum mechanics plays an essential role in the understanding of human consciousness, specifically that microtubules within neurons provide the brain with the hardware necessary to perform quantum computation and therefore that the collapse of the quantum wavefunction plays an important role in brain function.

Gödelian argument
In Shadows of the Mind, Penrose takes a new approach in arguing that consciousness is non-algorithmic, attempting a mathematical proof using Gödel's Incompleteness Theorem.

Gödel's Incompleteness Theorem
Gödel's Incompleteness Theorem can be expressed as follows:


 * Let X and Y be members of a data language D whose members can be used to represent executable programs that act on elements of D.
 * A program X applied to input data Y is written X(Y), and may or may not terminate in a finite number of steps.
 * Let pair be a function which maps D × D to some subset of D and is invertible. The function pair and its inverse should be computable. By an abuse of notation, abbreviate F(pair(X,Y)) as F(X,Y).
 * Let F be a function with the following property - if F(X,Y) terminates, then X(Y) does not terminate. Such an F is to be called consistent.
 * If the converse holds for such an F, i.e.
 * for all X and Y, X(Y) does not terminate => F(X,Y) terminates,
 * then we call F complete.
 * Gödel's theorem tells us that there is no F which is both consistent and complete. Proof:
 * Let F be consistent, and define G such that G(X)=F(X,X).
 * Then G(G) = F(G,G), and if G(G) terminates then this implies G(G) does not terminate.
 * Which is a contradiction, so we conclude that G(G) does not terminate.
 * Thus F is not complete. Q.E.D. Also F can be extended to a new F', which is in a sense more complete, with F' defined as F'(X,Y) = {if ((X=G) and (Y=G)) then terminate else F(X,Y)}.
 * The F' defined in the above proof is consistent, and it recognises a larger set of non-terminating programs, i.e. the set that F recognises, and G(G).

F can be interpreted as a theorem prover
We can regard F as representing a set of logical axioms and rules of proof in a mathematical theory that enables us to prove theorems about the non-termination of computer programs. Thus an execution of F(X,Y) is a "proof" of the theorem that X(Y) does not terminate. The system F is incomplete, in that it does not prove all statements about non-termination which are true, and F can be explicitly extended to prove a larger set of true statements. Of course the extended F is still incomplete (for exactly the same reason) and can itself be extended, and so on.

Interpreting F as a robot mathematician
We can regard F as an algorithm that specifies the operation of a robotic mathematician tasked with proving theorems about non-terminating programs. Gödel's Incompleteness theorem tells us that merely by inspecting the design of a such a robot, we can "know" something that the robot does not know, i.e. that G(G) as defined above is non-terminating.

Interpreting F as a description of the capabilities of a human mathematician
Let us make the following assumptions:
 * Human beings operate according to the laws of physics.
 * The laws of physics are computable, i.e. the behaviour of all physical systems can be predicted using algorithmic calculations.
 * From the behaviour of a human mathematician we can extract reliably correct theorems about non-terminating programs (e.g. when the mathematician writes a theorem down, submits it to a respected journal, and states when asked that they are "really, really sure" that the proof of their theorem is correct, then we assume the theorem is reliably correct).

From these assumptions we come to the conclusion that there is some algorithm F which is equivalent to the ability of a human mathematician to state correct theorems about non-terminating programs.

But we can derive G from F, as above, and know that G(G) is a non-terminating program.

But the reader is a human, so we have just proved that a human can know something that a human cannot know.

This is a contradiction, so one of the initial assumptions is wrong.

Penrose's conclusion is that the second assumption above is incorrect: certain aspects of the laws of physics, specifically those which govern the operation of human consciousness, are inherently non-computable, and therefore by extension human consciousness is likewise non-computable.

Experimental data seem to confirm at least this much of Penrose's conclusions. Bell's theorem, an inequality that should hold true if there are local hidden variables responsible for quantum indeterminacy, has been violated in the overwhelming majority of experiments designed to test it, indicating that there must be nondeterministic aspects to quantum mechanics.

Criticism
Penrose's views on the human thought process are not widely accepted in scientific circles (Drew McDermott, David Chalmers and others). According to Marvin Minsky, because people can construe false ideas to be factual, the process of thinking is not limited to formal logic. Further, AI programs can also conclude that false statements are true, so error is not unique to humans. Another dissenter, Charles Seife, has said, "Penrose, the Oxford mathematician famous for his work on tiling the plane with various shapes, is one of a handful of scientists who believe that the ephemeral nature of consciousness suggests a quantum process."

In May 1995 Stanford mathematician Solomon Feferman attacked Penrose's approach on multiple grounds, including the mathematical validity of his Gödelian argument and theoretical background. In 1996 Penrose offered a lengthy, consolidated reply to many of the criticisms of 'Shadows'.

Microtubule hypothesis
Penrose and Stuart Hameroff have constructed a theory in which human consciousness is the result of quantum gravity effects in microtubules. But Max Tegmark, in a paper in Physical Review E, calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in his support: "Physicists outside the fray, such as IBM's John Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior', he says." The Tegmark paper has been widely cited by critics of the Penrose-Hameroff proposal. It has been claimed by Hameroff to be based on a number of incorrect assumptions, but Tegmark in turn has argued that the critique is invalid.

Notes and references
This article includes text originally by Philip Dorrell which is licensed under the GFDL