John Allen Paulos reviews The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose, in his monthly Who’s Counting column at ABC News:
The first 400 pages of “The Road to Reality” sketch the mathematics needed to understand the physics of the following 700 pages. Like many mathematicians, Penrose is an avowed Platonist who believes that mathematical entities such as pi, infinite cardinal numbers, and the Mandelbrot set are simply “out there” and have an objective existence independent of us.
Developing his mathematical philosophy a bit with some interesting speculations about the relations between the mathematical, physical, and mental worlds (but never descending to sappy theology), he very soon gets into the mathematical nitty-gritty. He expounds on Dedekind cuts, conformal mappings, Riemann surfaces, Fourier transforms, Grassmann products, tensors, Lie algebras, symmetry groups, covariant derivatives, and fiber bundles among many other notions.
As suggested, the level of exposition and the topics covered make me wonder about the intended audience. Penrose writes that he’d like the book to be accessible to those who struggled with fractions in school, but this seems an almost psychotically optimistic hope. This is especially so because Penrose’s approach to so many topics is so clever and novel.
More here.