The Penrose–Lucas argument is a logical argument partially based on a theory developed by mathematician and logician Kurt Gödel. In 1931, he proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing Machine that works on Peano arithmetic because the latter can't see the truth value of its Gödel sentence, while human minds can. Mathematician Roger Penrose modified the argument in his first book on consciousness, The Emperor's New Mind (1989), where he used it to provide the basis of his theory of consciousness: orchestrated objective reduction.
Background
Gödel showed that any such theory also including a statement of its own consistency is 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.
An analogous statement has been used to show that humans are subject to the same limits as machines: “Lucas cannot consistently assert this formula”. In defense of Lucas, J. E. Martin and K. H. Engleman argued in The Mind's I Has Two Eyes that Lucas can recognise that the sentence is true, as there's a point of view from which he can understand how the sentence tricks him. From this point of view Lucas can appreciate that he can't assert the sentence-and consequently he can recognise its truth. Still, this criticism only works if we assume that we can replace Lucas' reasoning with a formal system that has a Gödel sentence, but the Penrose-Lucas argument tries to prove otherwise: our ability to understand this level of arithmetic is not a formal system that can be simulated in a Turing machine.
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 algorithm. Similar claims about the implications of Gödel's theorem were originally espoused by Turing in the late 1940s, by Gödel himself in his 1951 Gibbs lecture, by E. Nagel and J.R. Newman in 1958, and were subsequently popularized by the philosopher John Lucas of Merton College, Oxford in 1961.
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).