A Book With a Theory of Everything?

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.

Mathematical Preliminaries, Fractions to Fiber Bundles

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.)

The physics begins around page 400 and includes uncompromising discussions of space-time and Minkowskian geometry, general relativity theory of course, Lagrangian and Hamiltonian approaches to dynamics, quantum particles and entanglement including the standard illustrations (the two-slit experiment, Schrodinger's cat, and Einstein-Podolosky-Rosen non-locality), the measurement problem, Hermitian operators, black holes, the Big Bang, time travel, quantum field theory, the anthropic principle, Calabi-Yau spaces, as well as many other topics of current research.

EPR Experiment, String Theory, Inflation and Everything Else

As in his previous works the author is not afraid to strike an iconoclastic pose. He sides with Einstein and against most modern physicists, for example, in thinking that the EPR experiment demonstrates that quantum theory is incomplete.

The experiment, described very simplistically since this column has fewer words than Penrose's book has pages, involves identical particles moving rapidly apart. A physicist measures the spin of one of the particles realizing that quantum theory stipulates that the particle doesn't have a definite spin -- it could go either way -- until it is measured and its wave function collapses. Astonishingly, the other particle, which by the time of the measurement may be in a different galaxy, has a wave collapse at the same moment that always results in its having an opposite spin. How does the second particle instantaneously "know" the first particle's spin? Eerie entanglement, an incomplete theory, something else?

Penrose's skepticism extends to more modern developments as well.

He is unenthused about inflation theory and particularly so about string theory. (Inflation, very roughly, refers to the lightning fast expansion of a part of the very early universe, and string theory, even more roughly, refers to the notion that fundamental particles are composed of minuscule strings, vibrating and multi-dimensional.) Inflation theory has considerable evidence backing it, but Penrose seems correct to emphasize that string theory and its offspring M-theory are largely speculative. Why their appeal? He offers an interesting discussion of the role of fads and fashion even in theoretical physics.

The end of the book is devoted to a sketch of M-theory's main competitor, twistor theory and loop quantum gravity, which he invented decades ago and has been developing with colleagues ever since. He also seems less than impressed with Brian Green's "The Elegant Universe," a book that is far more accessible.

Coming every few pages, Penrose's well-done drawings and illustrations may ease the book's near vertical learning curve. Like some New Yorker subscribers, many readers of this book will, I suspect, confine themselves largely to the pictures and the pages that are more broad-gauged and less technical.

There is something to be said for inducing even this level of involvement in mathematics and physics, and if "The Road to Reality" succeeds in doing this, it may become a (sturdy) coffee table book and popular success. My hunch, however, is that this truly magisterial book will be appreciated primarily by those who have already spent considerable time in school learning a substantial portion of what's in it.

-- Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.