There's an oft-repeated story that when Stephen Hawking was writing "A Brief History of Time," he was told that every equation in the book would cut his readership in half.

If there were any truth to this counsel, Roger Penrose's "The Road to Reality: A Complete Guide to the Laws of the Universe," his recent 1,100 page behemoth of a book, should attract a half dozen readers at most. It's an enormous equation-packed excursion through modern mathematics and physics that attempts, quixotically perhaps, to answer and really explain "What Laws Govern Our Universe?"

Scattered about this impressive book are informal expository sections, but Penrose's focus is on the facts and theories of modern physics and the mathematical techniques needed to arrive at them. He doesn't skimp on the details, which, for different readers, is the book's strength and its weakness. Parts of it, in fact, seem closer in tone to a text in mathematical physics than to a book on popular science.

An emeritus professor at Oxford, Penrose is a mathematician and physicist renown for his work in many areas.

In the 1960s he and Hawking did seminal research on "singularities" and black holes in general relativity theory. He also discovered what have come to be called Penrose tiles, a pair of four-sided polygons, that can cover the plane in a non-periodic way. And about a decade ago he wrote "The Emperor's New Mind" in which he argued that "artificial intelligence" was a bit of a crock and that significant scientific advances would be needed before we could begin to understand consciousness.

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.

Another problem is that he doesn't generally proceed from the concrete to the theoretical, but more often in the other direction. (For the mathematicians: He introduces abstract 1-forms and only later the relatively more intuitive vector fields. Likewise he develops Maxwell equations via tensors and Hodge duals and never explicitly mentions more familiar notions like the curl of a field or Stokes' theorem.)

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