By: Roger Penrose, read in 2011

25 His description and analysis of the olive oil-water example is not convincing to me. It seems like energy has to be added by shaking or stirring which goes to breaking oil-oil bonds and water-water bonds. As this energy dissipates, the bonds are reformed. The "orbital velocity' story doesn't make sense to me.
27 "...we can describe the dynamical evolution of our system, according to these laws as a point p which moves along a curve—called an evolution curve in the phase space..." If p actually moves, that requires an additional temporal dimension—unless, of course the only thing that "moves" is the focus of attention of some observer of p.
27 "...the whole phase space...will be filled up (technically foliated) by such evolution curves (rather like a bale of straw), where every point of [the phase space] will lie on some particular evolution curve." I think it would resemble a bundle of steel wool rather than a bale of straw. Only by assuming the 2nd law are the strands forced to stay roughly parallel and proceed to regions of higher entropy. Big assumption IMHO.
50 It seems to me that his argument is weakened, almost to the point of vanishing, by examples like an egg which came from life processes. Time reversal of the egg, or any manufactured artifact, back to raw materials, show "entropy" going in the other direction. It seems that life is a special case that needs to play a bigger role in the definition of 'entropy'.
52 "...it has occasionally been argued that the existence of a Second Law holds no mystery, for our experience of the passage of time is dependent upon an increasing entropy as part of what constitutes our conscious feeling of the passage of time; so whatever time-direction we believe to be the 'future' must be that in which entropy increases." I disagree with this argument. Some of our most intensely experienced conscious periods are when we are creating some highly improbable artifact.
54 "...the mere existence of life provides, in itself, no argument whatever for the full validity of the Second Law." Exactly! In fact, IMHO it provides a counter-example.
78 "When animals eat plants (or eat other animals that eat plants), they use this source of low entropy, and the O2, to keep down their own entropy...'
256 "This general point seems to have been first made by Erwin Schrödinger in his remarkable 1944 book What is Life?"
88 Hyperbolic metric
90 Explanation of the negative terms in a Riemannian metric for time-like directions.
147 "I am suggesting that the universe as a whole is to be seen as an extended conformal manifold consisting of a (possibly infinite) succession of aeons, each appearing to be an entire expanding universe history."
149 "I refer to this cosmological scheme as conformal cyclic cosmology, abbreviated CCC." This is a picture of the physical universe. If you add to this Penrose's own two other "worlds", the Platonic and conscious worlds, you get something very similar to my own "Transfer of Omniscience", or the action of the double pendulum.
154 "A standard procedure for addressing the idea of an 'elementary particle' is to look for what are termed the 'irreducible representations of the Poincaré group'. Any elementary particle is supposed to be described according to such an irreducible representation. The Poincaré group is the mathematical structure describing the symmetries of Minkowsky space M, and this procedure is a natural one in the context of special relativity and quantum mechanics. The Poincaré group possesses two quantities referred to as Casimir operators, these being rest-mass and intrinsic spin, and accordingly the rest-mass and spin are deemed to be 'good quantum numbers', which remain constant so long as the particle is a stable one and does not interact with anything."
263 "These operators are the quantities constructible from the generators of the group which commute with all the group elements."
160 "...there turns out to be a clear-cut and apparently unique mathematical procedure for continuing the massless fields into the future through the crossover 3-surface, it being assumed that only massless fields are present in the very remote future of the earlier aeon..." In other words, when the physical world transitions from one aeon to the next, it consists only of radiation and the supporting field(s). ("Let there be light") This should be reviewed in the Nag Hammadi context.
178 "What will ultimately have happened to all this black-hole entropy?" I think the answer is the same as for what is going on on earth: The isolation and organization of some small part of the available energy by living and thinking beings reduces entropy locally in the face of increasing entropy globally (see "The Bottomless Well"). A thinking being might do the same on the largest scale leading up to the aeon boundaries. That is exactly consistent with my dentist chair experience recollection (double pendulum), and with Nag Hammadi revelations. It would create and provide the low entropy focal point for the next Big Bang. What ultimately happens to all the black-hole entropy is that it dissipates in the huge hyper-sphere surrounding the site of the next BB as illustrated in Penrose's Fig. 3.3, page 148.
184 A very nice description of the Mathematical account of Quantum Mechanics.
203 "It is clear...that there is no agreement whatsoever as to what might actually be going on in 'space-time' at the Planck scale."
203 Typo: 'were' should be 'where' second line from the bottom of the page
205 "...the detailed nature of the big bang of each aeon is completely determined by what happened in the remote future of the prior aeon" Consistent with my "Transfer of Omniscience"
256 Derivation, (or crude explanation) of the Minkowsky metric
266 "Schrödinger's equation is a complex first-order equation, and when time is reversed, the 'imaginary' number i must be replaced by –i"

Video of Roger Penrose explaining this idea: click here

Note to Roger Penrose:

I just finished and greatly enjoyed reading Cycles of Time. I would like to humbly, and respectfully, suggest that CCC be considered as a picture only of the physical universe, the domain to which Science enthusiastically and, so to speak, religiously, limits its inquiries. In order to get a more complete picture, I suggest that you add to CCC the other two of your three worlds as described on page 17 of The Road to Reality: The Mental World and the Platonic Mathematical World.

As you pointed out, these three worlds seem to exist in a paradoxical rock-paper-scissors causal relationship. To solve this causal paradox, I suggest considering one of these three worlds to be primary, or at least original. My preference for the choice would be the Mental World. Then following your causal cycle, some part of the Mental World would cause the Platonic Mathematical World, which in turn would cause the Physical World. Then, to avoid the paradox, the next cycle would be for the Physical World to give rise to a new and different Mental World. (I think of this schematically rising to a plane higher than that of the first 3 worlds, with a new consciousness hovering over the previous Mental World, so that a simple helix forms. Each successive turn of the helix gives rise to successively richer and more complex worlds. It seems natural to identify these successive levels with the aeons of CCC.

I have given this idea a lot of thought over the past 50 years or so and I would be delighted to discuss it further with you.

Paul R. Martin