These books often get shallowly dismissed in terms that imply he made some elementary error in his reasoning, but that's not the case. The dispute is more about the assumptions on which his argument rests, which go beyond mathematical axioms and include statements about the nature of human perception of mathematical truth. That makes it a philosophical debate more than a mathematical one.
Personally, I strongly agree with the non-mathematical assumptions he makes, and am therefore persuaded by his argument. It leads to a very different way of thinking about many aspects of maths, physics and computing than the one I acquired by default from my schooling. It's a perspective that I've become increasingly convinced by over the 30+ years since I first read his books, and one that I think acquires greater urgency as computing becomes an ever larger part of our lives.
* I will mention though that "some" should be "all" in 2, but that doesn't make it a correct statement of the argument.
I have since come to view it more as an interesting lesson in the pitfalls of hypothesis formation, popular non-fiction, and vanity.
Even so, as a layperson, it's entirely understandable to perk up whenever someone discovers 'tubules' in the brain, even if none of that sufficiently supports any of the collapse requirements of the Penrose/Hammeroff quantum microtubes.
"The Emperor's New Mind" is a great book on this and related topics.
- Godel, Escher, Bach - Hoffstadter
- The Soul of a New Machine - Kidder
- The Emperor's New Mind - Penrose
- The Connection Machine - Hillis
- Algorithmics - Harel