On the back of the book jacket it states that this book provides an "accessible" introduction to Roger Penrose's ideas. This is simply false, unless you are a physicist, a mathematician or a genius. For all others (in which category I fall), this book is close to incomprehensible.
Roger Penrose is brilliant, has a wonderfully clear mind, and can write beautifully and clearly. I think that _The Emperor's New Mind_ and _Shadows of the Mind_ are wonderful books. Penrose can make extremely difficult ideas wonderfully clear, by breaking down complex arguments into clear simple steps. Some pages of Emperor's and Shadows take a long time to understand, (I must admit that I have read quite a bit less than all of either book), but the argument is spelled out so clearly that with effort and patience it can be fully understood. Penrose, I think, understands the questions he is wrestling with as well as any one alive, and his is a mind of great depth as well as clarity. Even if you don't find his arguments entirely convincing, you can't help but learn a great deal from these books, and develop insights of great value. What went wrong with _The Large, the Small and the Human Mind_?
I think the key to answer lies in a comment Penrose makes on page 105, where he writes "The first 200 pages of _Shadows of the Mind_ were devoted to trying to show that there are no loop-holes in the argument that I am goind to give you." He then gives his argument in a few pages. The main part of the book (not including the replies of others and Penrose's replies to these replies) is just 139 rather small pages long. The book has many paragraphs like the following:
"The first is the Geroch-Hartle scheme for quantum gravity, which turns out to have a non-computable element, because it invokes a result, due to Markov, which asserts that topological 4-manifolds are not computationally classifiable. I shall not go into this technical matter, but it does show that this feature of non-computability has already come up in a natural way in attempts to combine General Relativity and quantum mechanics."
If, like me, you don't know what the Geroch-Hartle scheme is, and you don't know what topological4-manifolds are, you really won't know what to make of this paragraph. By "not going into this technical matter" Penrose is not omitting a minor technical detail of interest only to a select few, he is leaving readers like me completely baffled.
Penrose tried to write a short book that would give a clear idea of his basic ideas, and I think he failed. I know that he can make complex ideas wonderfully clear when he takes the time. In this book he has shown that, at least on this occaision, he can not write a book that is both brief and clear.
Here is another example, this time from Penrose's response to Abner Shimony's comments: (pages 176-7)
"Abner mentions Mielnik's non-Hilbertian quantum mechanics....In ordinary quantum mechanics, the space of density matrices consitutes a convex set, and the 'pure states', which would have a single state vector description, occur on the boundary of this set. This picture arises from an ordinary Hilbert space, being a subset of the tensor product of the Hilbert space and its comples conjugate (i.e. dual)."
Did you understand that? If so, you might learn a great deal from this book.
There are a number of topics that I have read about previously in other books, for example, non locality, Schroedinger's cat, the EPR experiment. In almost every case, Penrose's presentation was the worst I have seen. Probably everything he says is quite correct, but the presentation is so compressed that it was usually almost impossible for me to follow.
There are sections that I could follow, but even there he was so sketchy that I ended up with only a vague idea of _why_ he believes what he does, even when it was clear _what_ he believed. For instance, on pages 100-101 he is concerned with four approaches to awareness, which he labels A through D. "D" is as follows:
Awareness cannot be explained by physical, computational, or any other scientific terms.
Here is, in its entirety, what he has to say about "D":
Finally, the is always viewpoint D, according to which it is a mistake to look at these issues in terms of science at all. Perhaps awareness cannot be explained in scientific terms.
Penrose likes one of the other four positions, which is clearly stated, as is "D." But I would have liked to see Penrose develop in detail the reasons for his rejecting D. He does this only indirectly, by arguing for his preferred alternative, and this quite briefly.
I think that I understood enough of the book to say that if Penrose turns out to be even roughly correct, he will be regarded as one of the great geniuses of the century. His proposals are bold, and deal with questions of the greatest interest. If only I could have followed his arguments!
I hope that somebody who can follow his arguments posts a review.