Review: The Road to Reality, by Roger Penrose

Roger Penrose, The Road to Reality: A Complete Guide to the Laws of
the Universe
1136 pages including bibliography and index, Knopf (February 2005)

For a book ostensibly about reality, this one talks a lot about
“magic”, “mystery”, occasional “miracles”, and especially
the Platonic world of mathematics (or more generally, of “truth”), which
Penrose forcefully argues has a reality all its own. Those who
fear arithmetic may be lulled by Penrose’s gentle preface,
which goes into some detail on fractions.
This introduces the important mathematical principle of equivalence
classes; in the case of fractions, pairs of integers are declared
equivalent in a simple way that leads to the normal rules of rational
numbers.

But Penrose goes far beyond the simple in his discussion of the
mathematical concepts underlying modern physics theories.
Penrose’s perspective on the mathematics relevant to reality is
highly geometrical and visual, and even experts in the field might find
new insights in his discussion of hyperbolic geometries, complex numbers
(where lies much of the “magic”), and tensors and symmetry groups. Visual
insight is difficult when one is talking about higher dimensional
spaces, and the lack of visual geometric insight is one of
Penrose’s complaints about the 10 or more dimensions of typical
string theories. Penrose provides startling new ways to look at
the four dimensions most relevant to our spacetime, although occasionally
his descriptions are marred by poor diagrams. Some of the sections
also seem to be missing his unique touch, relying on
more pedantic and traditional approaches to the math.

For the very reasonable list price this is really three books in one,
and oddly organized for a book ostensibly directed at a lay audience
(it has been selling well!) First is a textbook on the mathematics
used by physical theories, complete with exercises. Penrose is
careful to define almost everything so an intelligent lay reader
could, in principle, follow, but the definitions
are brief and not entirely self-contained, and many readers will
lose hope fairly quickly. Second Penrose gives a semi-rigorous
introduction to the fundamental theories of modern physics, including the
remarkable successes of both General Relativity and the quantum field theory
of the standard model of particle physics. The real payoff is in the
final portion of the book, chapters 27-34, which describes the
vast gap that still remains between nature and our theories.

None of the other recent books along these lines – Hawking’s
“Brief History of Time”, Brian Greene’s books,
and other recent works by physicists touting our nearness
to a “theory of everything” have approached anything like this
level of detail. In Penrose’s view those other books are all at least
partially wrong, and he believes the mathematics needs to be
taken seriously to see why.

Penrose dives into this with an attack on inflationary
(and other) theories of the big bang, linking the thermodynamic concept
of “entropy” to the geometrical structure of the universe, where he is a
renowned expert. The fundamental issue, also discussed
in Hawking’s book, is why the universe has a definite
“arrow of time”, given that all our accepted physical theories are
symmetrical when time is reversed. Somehow the universe started in
a highly unusual low-entropy state; Penrose dismisses all the
usual arguments about how this could have happened with a
statistical analysis from his understanding of spacetime geometries.
Did the universe really require a creator to specially select the
astronomically low probability universe we started with, or are
there physical processes we don’t yet understand that introduce
this sort of radical time-asymmetry in a natural way? Penrose
favors the latter, but makes clear nobody has a good explanation.

The most substantive chapter in the book is Penrose’s discussion
of superstrings and related ideas. Some highly interesting
mathematics has come out of these studies, but
they may have little or no relation to the real world.
Superstring theories have so far predicted essentially nothing that can
be confirmed by experiment – Penrose explains why some often-touted
cases of prediction (such as black hole entropy) are in fact rather dubious.
Another part of Penrose’s problem with them, as
with the inflationary theories of the big bang, is that they are
undergoing continual rapid evolution so that the superstrings of
this year are very different from those of ten years back. Penrose
gives a reasonably unbiased presentation of the theories and why people
are interested in them, but he does not approve of the
bandwagon status they have acquired.

Penrose admits at various points that his views
on modern physical theories are somewhat out of the mainstream. His
major contributions to physics have ranged widely, but predominantly
within the realm of Einstein’s “General Relativity”, the theory
of gravitation. Two of his unique contributions that have not been
widely taken up by other physicists are mentioned repeatedly: his
view that gravitation is more fundamental than quantum mechanics, and
his “twistor” approach to spacetime geometry. The first of these implies
that the basic notions of quantum physics may need to be modified with
nonlinear terms arising from gravitational interactions within
quantum superpositions. For both he identifies new experiments and
theoretical work needed to make further progress, including
a space-based “Schrodinger’s Cat” experiment dubbed FELIX.

The fundamental thread through this book is this relationship between
mathematical theories and the reality of nature.
The Platonic world provides us with considerable “magic” to explore and
be fascinated by, but do the magic and occasional miracles lead us to
the truth about reality, or away from it? The reason mathematical truth
is not quite the useless tautology Wittgenstein once argued is
the linkage it provides between things we might have thought were
distinct, but are in fact the same from a certain perspective,
just as 3/6 and 1/2 are equivalent as fractions.
Penrose illustrates numerous examples of the way
mathematical equivalences can change one’s perspective on the nature
of the world around us – his “twistors” are yet another way of
looking at the world, beautiful but quite different from our usual
concepts of space and time. Sometimes in the past a beautiful
mathematical scheme, like that of the quaternions, has turned out to have
little practical use at all, despite much initial promise. Yet Einstein,
Dirac, Feynman and others were led by mathematical considerations to
theories that had real explanatory and predictive power. Penrose
emphasizes the importance of real experiments to guide physical theories.

It seems we simply don’t yet have enough experimental capability or
mathematical understanding to find the true “theory of everything”.
It’s possible we’re close; if we are this book may help to make that
happen. But even if we still have a long and winding
road to follow, from Penrose’s perspective there will be much
that is fascinating and beautiful to observe along the way.