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Mon, 04 Jul 2005

Penrose, Road to Reality

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

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

Laura Conrad

Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Knopf
Roger Penrose states in the preface of this book, "The purpose of this book is to convey to the reader some feeling for what is surely one of the most important and exciting voyages of discovery that humanity has embarked upon." He then procedes to justify his inclusion of mathematical formulas, but to advise the reader, "Do not be afraid to skip equations (I do this frequently myself) and, if you wish, whole chapters or parts of chapters, when they begin to gget a mite too turgid!" He also says, "My hope is that the extensive cross-referencing may suficiently illuminate unfamiliar notions, so it should be possible to track down needed concepts and notation by turning back to earlier unread sections for clarification."
The first thing you may notice about this description of how to read the book is that it isn't much like bedtime reading. For a book to be good bedtime reading, it needs to meet several conditions:
  • You must physically be able to see and turn the pages from a recumbant or semi-recumbant position. "The Road to Reality", at 1000+ pages and about 4 pounds, makes this difficult but not impossible.
  • The writing needs to be modular enough that you can imbibe some satisfactory idea in the time between getting into bed and turning out the light to go to sleep. It is conceivable that the kind of browsing that Penrose recommends would turn up some sections that would work like this, but I found that trying to read the first few chapters straight through didn't do this for me.
  • Understanding the work must not involve extensive use of paper and pencil to work out problems, indexes, references to other books, and other study methods more appropriate to sitting at a desk. I have not attempted to work the exercises. But I found that almost all the writing in this book was dense enough to require reference to other parts of the book, often separated by many pages or chapters and findable only by using the index.
This book is being marketed by Knopf as being aimed at the popular science market, presumably the people who read Scientific American and Steven Hawking. I found it much harder reading than those books.
For a great deal of my life I've been an enthusiastic consumer of the literature of popular science. In elementary school, I read and reread the book about the solar system by Patrick Moore. In high school I devoured the books about cosmology, stellar evolution, and quantum physics by George Gamov and Fred Hoyle. I was led by the picture of what physicists do I derived from these books to make one of the major mistakes of my life and major in physics in college. One book that allowed me to survive my four years as a physics major was "The Feynman Lectures in Physics" by Feynman, Leighton and Sands. This was not marketed as popular science writing, but Penrose is claiming to write on something like the level of this book, which is a heavily edited version of the lectures Richard Feynman gave to a freshman physics class at CalTech.
I did in fact read the Feynman lectures in bed as an undergraduate, although I also did the reading at a desk with index and comparisons to my textbook and working out problems with pencil and paper. Since I was finding Penrose's writing much harder than I remembered Feynman's, I decided to compare them on the same topic.
Here's the first mention of the term "partial derivative" from the index of both books. In Feynman, it's in chapter 14, "Work and Potential Energy (conclusion)":
We find that the force is:
F=-ΔU/Δx
Of course this is not exact. What we really want is the limit as Δx gets smaller and smaller, because it is only exactly right in the limit ofinfinitesimal Δx. This we recognize as the derivative of U with respect to x, and we would be inclined, therefore to write -dU/dx. But U depends on x, y, and z, and the mathematicians have invented a different symbol to remind us to be very careful when we are differentiating such a function, so as to remember that we are considering that only x varies, and y, and z do not vary. Instead of a d they simply make a "backwards 6", or . (A should have been used in the beginning of calculus because we always want to cancel that d, but we never want to cancel a !) So they write U/x, and furthermore, in moments of duress, if they want to be very careful, they put a line beside it with a little yz at the bottom ( U/x |yx ), which means "Take the derivative of U with respect to x, keeping y and z constant." Most often we leave out the remark about what is kept constant because it is usually evident from the context, so we usually do not use the line with the y and z. However, always use a instead of a d as a warning that it is a derivative with some other variables kept constant. This is called a partial derivative; it is a derivative in which we vary only x.
And in Penrose, it's in Chapter 10, "Surfaces":
I have not yet explained what 'differentiability' is to mean for a function of more than one variable. Although intuitively clear, the precise definition is a little too technical for me to go into thoroughly here. Some clarifying comments are nevertheless appropriate.
First of all, for f be [sic] differentiable, as a function of the pair of variables (x,y), it is certainly necessary that if we consider f(x,y) in its capacity as a function of only the one variable x, where y is held to some constant value, then this function must be smooth (at least C1 ), as a function of x, in the sense of functions of a single variable (see §6.3); moreover, if we consider f(x,y) as a function of just the one variable y, where it is x that is now to be held constant, then it must be smooth ( C1 ) as a function of y. However, this is far from sufficient. There are many functions f(x,y) which are separately smooth in x and in y, but for which would be quite unreasonable to call smooth in the pair(x,y). A sufficient additional requirement for smoothness is that the derivatives with respect to x and y separately are each continuous functions of the pair (x,y). Similar statements ... would hold if we consider functions of more than two variables. We use the 'partial derivative' symbol to denote differentiation with respect to one variable, holding the other(s) fixed. The partial derivatives of f(x,y) with respect to x and with respect to y, respectively, are written
f / x and f / y
(As an example, we note that if f(x,y)= x2 + xy2 + y3 , then f/x=2x+ y2 and f/y=2xy+3 y2 .) If these quantities exist and are continuous, then we say that Φ is a ( C1 -) smooth function on the surface.
Note that this appears on page 183-4, the definitions of Cn smoothness are on page 110. You can find this from the index.
My guess is that if you don't already have some idea of why you want to know what a partial derivative is, you won't make it through either quote very easily. But I think you'd have a lot better chance of figuring out what it is from the Feynman explanation.
I was hoping when I read about the Penrose book that it would be as interesting and well-written as I had found the Feynman book, but have the stuff that's been done in the forty years since the Feynman book was written. It does indeed have some of that stuff, but I think the reader will have to be very determined indeed to get to it. I don't see any chance at all of someone who isn't already excited about modern physics becoming so by reading this book.



File translated from TEX by TTM, version 3.68.
On 30 Jun 2005, 18:07.

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