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 von Arnim, Solitary Summer

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
crossreferencing 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 semirecumbant 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:
$F=\Delta U/\Delta x$
Of course this is not exact. What we really want is the limit as
$\Delta x$ gets smaller and smaller, because it is only exactly
right in
the limit ofinfinitesimal
$\Delta x$. This we recognize as the derivative of
$U$ with respect to
$x$, and we would be inclined,
therefore to write
$\mathrm{dU}/\mathrm{dx}$. 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
$\partial $. (A
$\partial $ should have been used in the beginning of
calculus because we always want to cancel that
$d$, but we
never want to cancel a
$\partial $!) So they write
$\partial U/\partial x$, and furthermore, in moments of duress, if they want to be
very careful, they put a line beside it with a little
$\mathrm{yz}$
at the bottom (
$\partial U/\partial x{}_{\mathrm{yx}}$), 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
$\partial $
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
${C}^{1}$), 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 (
${C}^{1}$) 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
$\mathrm{pair}(x,y)$. 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
$\partial $ 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
$\frac{\partial f}{/}\partial x$ and
$\frac{\partial f}{/}\partial y$
(As an example, we note that if
$f(x,y)={x}^{2}+{\mathrm{xy}}^{2}+{y}^{3}$,
then
$\partial f/\partial x=2x+{y}^{2}$ and
$\partial f/\partial y=2\mathrm{xy}+3{y}^{2}$.) If these quantities exist and are continuous, then we
say that
$\Phi $ is a (
${C}^{1}$) smooth function on the surface.
Note that this appears on page 1834, the definitions of
${C}^{n}$
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 wellwritten 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
T_{E}X
by
T_{T}M,
version 3.68. On 30 Jun 2005, 18:07.
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