The Road to Reality: A Complete Guide to the Laws of the Universe Reprint Edition

This is an extraordinary book and certainly there is nothing else available on the market that is quite like it.

Penrose is a pre-eminent mathematician whose work has affected my own thinking in deep ways. Often I've discovered, to my surprise, that some viewpoint which I learned from another source in fact derived ultimately from Penrose himself, the author of this other source having been, as it turns out, a student of Penrose. As an example I offer "Visual Complex Analysis" by Tristan Needham. I encountered this book in high school and so powerful was its exposition that it stuck with me all through college and beyond to the present day. I couldn't help but notice, while reading "The Road to Reality," that Penrose engaged in a similar revelry around complex numbers -- and so it was a pleasant surprise to see that Needham indeed had studied directly under Penrose! Despite having this book in my collection for a few years only recently did I discover the connection.

With this tangential introduction out of the way, let me say at the outset that I cannot recommend this book to just anyone. It requires a great deal of mathematical acumen -- or perhaps, more accurately, a great deal of explicitly mathematical *interest*. You needn't be a mathematician, but you positively must delight in the opportunity of learning the mathematics of the universe. Even if you have a degree in mathematics you're likely to find something here you haven't encountered before -- for instance, I had not previously seen the notion of a hyperfunction which Penrose enthusiastically presents in connection with the Fourier transform.

The book's mathematical explanations are quite idiosyncratic, and I think this reflects the depth of Penrose's intuition on the topics. It elevates this book from an encyclopedic reference to a personal outpouring of the author's vision of the world. Not only does the book feature unique topics that aren't commonly taught, the aforementioned hyperfunctions being an example, but even those which are commonly taught are explicated here in a unique manner. Penrose's style is unabashedly geometric, and so even partial differential operators on manifolds he almost always thinks of as vector fields. The book is filled with many beautiful hand-drawn diagrams showing everything from a fanciful vision of the creation of the cosmos to a field of one-forms on a surface to the conservation of electric charge in spacetime. And the book prominently features throughout his diagrammatic notation for tensor algebra, something again which I had never encountered elsewhere.

I should say that this book approaches its physical topics from an exceptionally high level. You will probably not be able to learn electrodynamics from Penrose, not here. I don't recall the common form of Maxwell's equations even occurring in this book. Rather, he simply writes them down in the simplest, most beautiful, and most abstract form, using differential forms and exterior calculus. To delve in deeper, rather than expanding out these equations, he instead focuses on how they can arise as the curvature of a bundle connection via gauge theory. This is both stunningly beautiful and horrendously detached from practice.

Despite the overwhelming emphasis on theory and mathematics, the book is quite adamant in insisting that physics still must be physics. Penrose is not optimistic about string theory, though he does give it some treatment towards the end for completeness. He's perhaps like Feynman in that regard -- he possess a deep appreciation for the mathematics and the theory, but at the same time is very uneasy with the lack of testability of much of modern theoretical physics. His own theory, twistor theory, is just as untestable, though he's honest about that.

With just under 1100 pages, this is no light-weight book, and the content matches. Penrose is an extremely capable mathematician with deep physical understanding, and it shows. It is not, however, suitable for readers with little mathematical or physical understanding. It is not a book for basic learning, but rather it seems directed at a very limited audience with sufficient background knowledge and for them it will take them far further. The book is designed to take people with advanced physical and mathematical knowledge up to Penrose's level of understanding. The first third of the book is effectively an introduction to advanced mathematics, and is interesting for this in its own right.

The book nominally takes the reader with no background information and fills this in, but I do not believe that is plausible. For example, consider the introduction to Maxwell's equations. These are introduced in Minkowski coordinates and expressed into a spacetime 2-form F, called the Minkowski field tensor, and a further space-time vector, with components expressed in matrix form that permit various symmetries to be perceived. I do not believe that anyone who had not at least vaguely become aware of this sort of approach would make much headway, but what follows is certainly elegant.

My biggest "grumble" with this book is the title: I believe "The" should be replaced with "A". The ancient Greeks founded two approaches to physics. Penrose adopts the Platonic approach, which considers mathematical forms, symmetries, and even beauty, and tries to fit reality to mathematical form. Thus towards the end, Penrose devotes a chapter to twistor theory, and in the end has to acknowledge that this is mathematically attractive, but it is difficult to take it anywhere in physics. The alternative approach, the Aristotelian approach, asks questions of nature and tries to find restrictions that lead to the correct theory. I should mention the caveat that I fall into the latter school, and with that warning, I believe that a true route to reality will involve many people working from both approaches.

Irrespective of certain reservations, I believe that anybody with some mathematical ability will gain some insights from this book (the more the ability, the more insights), and anybody interested in theory should at the very least read chapter 34, which gives Penrose's opinions on where the road will go, and the fashions encountered on it. This is a truly great piece of intellectual effort.

Ian J Miller, author of Elements of Theory