The Penrose–Lucas argument is a logical argument partially based on Kurt Gödel's first incompleteness theorem. In 1931, Gödel proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. John Lucas and Roger Penrose postulate that this incompleteness does not apply to humans, and conclude that humans can have mathematical insights that Turing machines can't. Penrose and Stuart Hameroff proposed a quantum explanation, and used it to provide the basis of their theory of consciousness: orchestrated objective reduction. The argument is rejected by most scholars.
Background Gödel's first incompleteness theorem shows that for any consistent formal system F {\displaystyle F} that allows certain arithmetic operations, there are statements of the language F {\displaystyle F} that cannot be proved or disproved. Thus, F {\displaystyle F} is either incomplete or inconsistent. A key element of the proof is the use of Gödel numbering to construct a "Gödel sentence" for the theory, which encodes a statement of its own incompleteness: "This theory can't prove this statement"; or "I am not provable in this system". Either this statement and its negation are both unprovable (the theory is incomplete) or both provable (the theory is inconsistent). In the first eventuality the statement is intuitively true (since it is not provable); otherwise, the statement is intuitively false - though provable. According to proponents of the Penrose–Lucas argument, there is a disjunction: either the human mind is not a computation of a Turing machine, and thus not an effective procedure; or it is a product of an inconsistent Turing Machine that could be reasoning using some sort of paraconsistent logic. Gödel himself commented about this disjunction in 1953. Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as formal proof systems (which theorems can be proved using an abstract object such as a computer), and are therefore running a non-computable process. The argument was originally considered and dismissed by Turing in the late 1940s. It was espoused by Gödel himself in his 1951 Gibbs lecture, by E. Nagel and J.R. Newman in 1958, and was subsequently popularized by the philosopher John Lucas of Merton College, Oxford in 1961.
The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation!
Consequences If correct, the Penrose–Lucas argument creates a need to understand the physical basis of non-computable behaviour in the brain. Most physical laws are computable, and thus algorithmic. However, Penrose determined that wave function collapse was a prime candidate for a non-computable process. In quantum mechanics, particles are treated differently from the objects of classical mechanics. Particles are described by wave functions that evolve according to the Schrödinger equation. Non-stationary wave functions are linear combinations of the eigenstates of the system, a phenomenon described by the superposition principle. When a quantum system interacts with a classical system—i.e. when an observable is measured—the system appears to collapse to a random eigenstate of that observable from a classical vantage point. If collapse is truly random, then no process or algorithm can deterministically predict its outcome. This provided Penrose with a candidate for the physical basis of the non-computable process that he hypothesized to exist in the brain. However, he disliked the random nature of environmentally induced collapse, as randomness was not a promising basis for mathematical understanding. Penrose proposed that isolated systems may still undergo a new form of wave function collapse, which he called objective reduction (OR). Penrose sought to reconcile general relativity and quantum theory using his own ideas about the possible structure of spacetime. He suggested that spacetime is not continuous at the Planck scale, but discrete. Penrose postulated that each separated quantum superposition has its own piece of spacetime curvature, a blister in spacetime. Penrose suggests that gravity exerts a force on these spacetime blisters, which become unstable above the Planck scale of 10 − 35 m {\displaystyle 10^{-35}{\text{m}}} and collapse to just one of the possible states. The rough threshold for OR is given by Penrose's indeterminacy principle:
τ ≈ ℏ / E G {\displaystyle \tau \approx \hbar /E_{G}}
where:
τ {\displaystyle \tau } is the time until OR occurs,
E G {\displaystyle E_{G}} is the gravitational self-energy or the degree of spacetime separation given by the superpositioned mass, and
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