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 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 2^{2n}+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 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, employes 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.

(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.

(page 620) **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.

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.

A very interesting review