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I'm hoping that this board may be able to offer an explanation. First, my apologies for quoting a slightly lengthy fragment but I wanted to provide the full context for my question. I'm reading Artificial Intelligence - A Modern Approach1 where I came across the following assertion:

26.1.2 The mathematical objection

It is well known, through the work of Turing (1936) and Gödel (1931), that certain mathematical questions are in principle unanswerable by particular formal systems. Gödel’s incompleteness theorem (see Section 9.5) is the most famous example of this. Briefly, for any formal axiomatic system F powerful enough to do arithmetic, it is possible to construct a so-called Gödel sentence $G(F)$ with the following properties:

  • $G(F)$ is a sentence of $F$, but cannot be proved within $F$.
  • If $F$ is consistent, then $G(F)$ is true.

Philosophers such as J. R. Lucas (1961) have claimed that this theorem shows that machines are mentally inferior to humans, because machines are formal systems that are limited by the incompleteness theorem—they cannot establish the truth of their own Gödel sentence—while humans have no such limitation. This claim has caused decades of controversy, spawning a vast literature, including two books by the mathematician Sir Roger Penrose (1989, 1994) that repeat the claim with some fresh twists (such as the hypothesis that humans are different because their brains operate by quantum gravity). We will examine only three of the problems with the claim. (...)

Question

I'm particularly intrigued by the highlighted statement. Unfortunately, the book doesn't elaborate on the significance of Penrose's work. Could someone explain Penrose's claim to me?

1Russell, S. J., & Norvig, P. (2016). Artificial intelligence: A modern approach. Worldcat, Goodreads.

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The highlighted text refers to quite controversial hypothesis of Orchestrated objective reduction or Orch OR by R. Penrose & S. Hameroff. Its preliminary formulation could be found in two poular scince books by Penrose: The Emperor's New Mind (1989) and Shadows of the Mind (1994) while a (relatively) recent review and responses to criticisms could be found here:

  • Hameroff, S., & Penrose, R. (2014). Consciousness in the universe: A review of the ‘Orch OR’ theory. Physics of life reviews, 11(1), 39-78, doi:10.1016/j.plrev.2013.08.002.

In short, quantum gravity causes wavefunctions' collapse, generally at random, but in brain microtubules this occurs on purpose and the process is responsible for computations and consciousness.

Wikipedia links to some of the many criticisms of the hypothesis as well as to authors' responses to criticisms.

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