Penrose proves to himself that our consciousness is non-algorithmic, and that we seem - to our conscious selves - able to make decisions in a flash.
This to him is strange, and he speculates why. He finds that this could be explained by quantum physical thought processes that proceed in sub-graviton parallellism until they reach graviton level, when they collapse and produce a concsious thought.
Her view is strongly grounded in biology. She proposes a physical explanation of biological functions: she believes micro-wave transmissions that "align" cells to each other over distance explain how non-local phenomena may occur on a superatomic level.
A notable case is that the retina accepts photons, which are small enough to behave strangely in terms of classical physics.
Quantum physics is strange. So is consciousness. Maybe there is something in common between the two.
He goes on to claim that mind without biology is a non-starter. But neither Penrose nor Zohar claim mind without biology! They try to explain biology.
Some processes in quantum physics are indeterminate until measurements are made on them. Several different outcomes are possible, and the result needs not be determined before the outcome concerns particles on a larger scale.
The indeterminism is commonly modeled in a wave function - which is a combination function of possible outcomes, and determining the outcome is commonly termed "collapse of a wave function".
Picture from Zohar, page 41.
The cat: "The consciousness of the observer has killed it!" Zohar points out that this is silly. If there are paradoxes in the application of quantum theory, this has to do with quantum theory, not with consciousness. Mend the theory, she says.
Crystals - picture from Penrose, page 436.
Magnets - picture from Zohar, page 83.
Nervous systems as electric circuitry - picture from Penrose, page 390.
Twin particles - picture from Penrose, page 282.
This scepticism is entirely based on introspection and awareness of conscious thought, and thus cannot be taken for more than a feeling. And as can be seen from most materials on the subject, the arguments range from "it is obvious" to "this is what once happened to a colleague of mine".
Typically, formal axiomatic systems such as formalizations of Turing machines assume that an algorithm operates on data expressed using symbols from a finite alphabet and that the data processed can be infinitely large.
The critique against algorithmic models of thought generally focuses on one obvious oversimplification in axiomatizing information processing.
Can you see which one?
It is patently obvious that the alphabet human information processors make us of is not finite. We coin new words at every turn of a clause, and we use old words in novel ways at the least opportunity.
Besides, it is not clear what the words mean, even when they are old and well worn.
Various forms of this critique are known as "the symbol grounding problem" or the "chinese room".
Human information processing never operates on infinitely large data. Sentences are never infinitely long. Clauses are never infinitely complex. In fact, it is theoretically possible to list all clauses that a human listener or reader can understand. (cf. Boman and Karlgren, 1996).
This is ridiculous.
Thus, we can say that no, the brain is no Turing machine - not because we need to prove its non-algorithmic nature, but simpler, because the formalism used is simply not appropriate for modeling the brain's behavior.
The algorithmic nature of thought is not proven, nor is it disproven by virtue of Turing excercises.
Penrose's main point is that our thinking is non-algorithmic.
Penrose says that consciousness as a side effect of running an algorithm is not possible. (pp. 405-7)
Penrose also believes that pure incomputability is not strong enough as an explanation for the perceived non-algorithmic nature of thought (pp 431-432): "However, it is very hard to believe that this kind of classical uncertainty can be what allows us our (illusion of?) free will."
But there is ample evidence that consciousness is a desirable mechanism in itself, and that conciousness is a tool, developed for a purpose. Neither of the authors address this; they see consciousness as a side effect of intuition, awareness, or of existence. Penrose discusses at length why natural selection would have developed consciousness if it weren't for some use (pp 408-9) but never proposes any use for consciousness apart as being a contrast to intuitive thinking.
If we see consciousness not as a side effect of algorithm operation nor of awareness or existence, but as a tool developed for a purpose, such as would Humphries (1980).