I wrote the following mostly in 2006 as I began to read the book. It is a big book and I sometimes come back to it and read more. I will add years in parens for updates.
I am beginning Penrose’s The Road to Reality.
Penrose gallops furiously thru the concepts. He covers what was a difficult semester course in a few pages. Yet I have learned some important math and physics from reading Penrose elsewhere and so I follow his arguments closely here. So far (page 209) I follow mainly because I have been there before but I fear he will not slow down when he exceeds my familiar ground. Perhaps he will at least identify and name fundamental concepts necessary for some modern physics. The book may be more of a map of the road to reality than a conveyance thereon, but such a map is much needed.
(page 18) I agree with Penrose’s categories of existence and I am of his latter category of mathematicians. Les Lamport said it best: “I am of the old school that believes that a theorem either does or does not follow from some particular set of axioms.”. I do not separate the mental world from the physical world. I can imagine a set of computers seeking to prove theorems long after all biological entities have ceased to exist. They would continue to advance mathematics by discovering new platonic truths with no mentality involved.
I would not object to carving out a sort of physical activity and calling it mental and then arguing where we should draw the boundary. That boundary might or might not include what computers do, depending on taste. Those arguments bear on the scope of a word, not what thought consists of.
(page 20) Penrose allows that there may be mathematical truths beyond human understanding, but he thinks not. I have disagreed with Penrose in The Emperor’s New Mind. I suspect that there is neither a formal nor an intuitive proof that there are only 5 Fermat primes (primes of the form 22n+1), yet I think it is true but that it is true for no good reason. Certainly there are such propositions that are true with being provable with our conventional axioms but Penrose thinks we will find new convincing axioms with which to prove the true ones. I doubt that.
(page 21) Quote from a section that he introduces as contentious: “… in my opinion, there is little chance that any deep understanding of the nature of the mind can come about without our first learning the very basis of physical reality.”. My prejudice is the opposite: clocked deterministic Boolean logic entirely suffices.
(page 56) I have learned a few things about Greek mathematics here. Penrose cites a theorem by Lagrange that quadratic irrationals ((a/b)+√(c/d)) have repeating continued fractions, and vice versa. Wolfram agrees with Penrose. I had thought you could get solutions to 3rd degree polynomials from repeating continued fractions—seems not.
(page 122) Penrose introduces the derivative of a complex function. He should have defined the derivative by saying that the finite slope value (f(x+Δx) − f(x))/Δx must converge to the same s no matter how Δx approaches 0, then f '(x) = s. One can then conclude from this definition that |x| has no derivative for 0 or non real x. Carrying this concept of derivative over to the complex domain would automatically produce the Riemann-Cauchy equations rather than the somewhat obscure rationale he gives on page 194. You get more results in the complex plane from requiring such a derivative because there are many more ways for Δx to tend to zero. In the reals there are two sides of 0, in the complex plane there is a whole circle of sides. For instance suppose f(x) = x*, the complex conjugate function. f(x) is smooth, but has no derivative since the finite slope value is 1 for real Δx but −1 for imaginary Δx.
(page 172) I like the development of hyperfunctions to which Dirac’s delta function belongs. These are new to me. He starts with functions of the sort that Euler would have accepted, employees Fourier’s discoveries, and ends up with a class of functions that probably make most of today’s mathematicians queasy. It all appears to be above board however. The Laurent series plays a critical rôle. I think you can re-jiggle Hilbert space to include hyperfunctions.
(page 185) When Penrose introduces 1-forms he uses conventional physics notations where we write “f(x)” as a function instead of “f”. This causes confusion, but so do other notations. He describes the confusion well.
(page 208) I really like his introduction to Clifford algebras. I could not get traction on the subject from Hestenes’ book on the subject. Now I know a short definition as well as a purpose for them. I am tempted to try a more complete introduction in the sense of filling in more details. Here are some resulting notes on Clifford algebras.
(2015) I have written many pages now on Clifford algebras. Thanks Penrose.
(page 219) Penrose describes how a loop thru orientation space may be unshrinkable, indicating that the space is not simply connected. I had previously described that space but failed to make the elementary observation that the space was not simply connected. I sort of missed the punch line.
(page 222) Penrose requires coordinate patches to be consistent when they overlap by threes. Conventional definitions do not require this. It seems obvious that Penrose is right. In n dimensions n+1 coordinate patches necessarily overlap at some point. I suppose that if they agree by threes they all agree by transitivity.
(page 241) Diagrammatic Notation
I immensely enjoy Penrose’s diagrammatic notation. I once took small steps in this direction but nothing nearly so complete and flexible. There remains the problem of typography.
Penrose’s introduction to Lagrangians and Hamiltonions (page 471) got me off on a wild goose chase after the ‘right definition’ of the group of a fiber bundle. It seems that there are a variety of such definitions for different purposes.
(page 620) (2008) Dirac’s Equation for the Electron
I am very pleased to see Penrose’s Clifford version of Dirac’s relativistic solution for the electron. I now know something about associative algebras, thanks partly to Penrose. I think about that equation here.
(page 675) (2015) I do not like Penrose’s explanation of renormalization, but it is better than any other I recall.
(page 738) (2015) I find myself blocked in chapter 28 by vital references to chapter 26 which I had skipped—back to 26!
(page 1030) (Yes 1030 is a page number, not a location. This is a big book.) This note is sort of a confession on where I stand. Penrose raises the rôle of mentality in our understanding of the universe. I trust Penrose’s views on physics more than anyone but perhaps Feynmann or Einstein. I have little regard for his views on QM and the brain however. The questions that Penrose addresses here are understood only in light of what our brain evolved to achieve—to keep us alive, Evolutionary Epistemology. (. .) These issues are founded on our DNA but we don’t know how yet.
A mere nit perhaps: I think we could do physics without consciousness but perhaps not as well. The disagreement here is the scope of the word consciousness perhaps. In this quote “… since consciousness is precisely what plays the ultimate role of observer.” Penrose proposes a reasonable partial definition of consciousness. I take it that to be conscious is to observe. Predators observe their prey but I would not ascribe consciousness to all predators.
(page 1031) I accuse Penrose of a vaguely dualist perspective here. “In fact almost all of the ‘conventional’ interpretations of quantum mechanics ultimately depend upon the presence of the ‘perceiving being’, and therefore seem to require that we know what a perceiving being actually is!” I thought that the first page of Everrett’s paper made it clear that an observer, in his formalism, was no more and a molecule that was somehow impacted by an uncertain event. There is no more to observation than to be impacted. Us animals, especially humans, have built models of the world which we call beliefs. No difference. I think that Penrose, as observer, wants to stand back on safe side of the wave collapse. No such luck—we are all in it together.
Later on page 1031: “The many-worlds viewpoint is manifestly dependent upon having a proper understanding of what constitutes a ‘conscious observer’, since each perceived ‘reality’ is associated with an ‘observer state’, so we do not know what reality states (i.e.‘worlds’) are allowed until we know what observer states are allowed.”
I take Everett’s proposal to be much more radical; every atom is an observer. There is no reality but the configuration space for the entire universe and a unitary map of measures thereon to itself. There is a real need of what constitutes a world model. AI engineers need to know. I do not buy much of §34.7.