My Goodreads Review
rating: 3 of 5 stars

First, the good. Penrose weaves tales of science, philosophy, and history that few others can, due to his wide-ranging and vast intellect. He touches on a wealth of interesting subjects in this book and his enthusiasm for them bleeds through the pages. In particular, this book offered the most illuminating introduction to entropy (in the “Cosmology and the arrow of time” chapter) that I have ever read. In short, before I read it, I didn’t believe in the second law of thermodynamics. After I read it, I could interpret the entire world through “entropy goggles”. In other books, entropy was some esoteric concept; here it is a beautiful and central feature of our universe.

Now, the irritating. Penrose’s stated goal in this book is to convince the reader that strong AI will not be realized. In other words, what human brains do, no computer ever could. Yet, however many fascinating ideas he introduces, his core evidence is little more than hope that what he does as a mathematician is unique; that human “aha!” moments are somehow distinguished from anything a computer ever does.

Penrose fervently declares that he is no “formalist”, claiming that the mere idea of a computer arriving at mathematical proofs makes the pursuit of mathematical truth “meaningless”. Why? A computer may arrive at such a truth, but it is the human’s role to make human meaning out of it. Meaning isn’t some quality embedded into the fabric of our universe. It is a manufactured human ideal - a wonderful, enjoyable one at that.

Penrose goes on to cite the vague notion of “reflection principles” (the imprecise methods by which the human mind discovers mathematical truth “upon reflection”) as above and beyond algorithms. He uses these “aha!” moments as support for his argument that algorithms cannot possibly imitate what we do. Yet, how can we be so sure that these “aha!” moments do not arise from some complicated algorithm themselves?

Perhaps, any human-created set of mathematical axioms will always be subject to “reflection principles” and only a more intelligent species could create a set for which there are no human-generated “reflection principles”. Perhaps the notion of computability simply refers to a hierarchy of skill among complex adaptive systems. (If anyone knows of a book or research on such an idea, please let me know)

Finally, on Penrose in general. Penrose’s books tend to occupy a literary no-man’s-land between popular science and technical writing: too technical for the average hobbyist yet not deep enough for a student in the sciences. It seems he’s just too damn smart to recognize what falls into each category (for example, he might spend half a chapter explaining fractions and then breeze through Hamiltonians in half a paragraph). For me, he’s best as a connector of mathematical and physical ideas after I’ve been formally introduced to them.

As any good book should do, this one did leave me with a few questions (besides the above):

• Which is a deeper truth - math or physics?
• In reflecting on the story non-Euclidean geometry, what else are our evolutionary adaptations leading us to falsely assume?
• Has anyone checked whether Hamiltonians would be deterministic for computable universal constants and discrete input?
• Can we really separate dynamical equations from boundary conditions? Are not boundary conditions simply consequences of other not-yet-understood dynamical equations?

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