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Five More Golden Rules by John Casti, Wiley, £18.50, ISBN
0471322334

AN APOCRYPHAL tale has it that Queen Victoria was so taken with Lewis
Carroll’s Alice stories that she insisted on being sent a copy of his
next book. Keen to avoid incurring the monarch’s wrath, Carroll—aka
Charles Dodgson, an Oxford maths don—did as he was instructed, and duly
sent Her Majesty an impenetrable treatise on determinants.

The story may be bunk, but I was beginning to know how Queen Victoria might
have felt within an hour of starting to read John Casti’s Five More Golden
Rules.

As the title suggests, Casti’s new book is a follow-up to his outstanding
Five Golden Rules, which I reviewed for 91av (20
January 1996, p 42). In that book he brought a quintet of key theorems of
20th-century mathematics to a wider audience. His sequel aims at showcasing five
more outstanding ideas from modern mathematics. Sadly, it bears the same
relationship to its predecessor as that treatise on determinants had to
Alice in Wonderland.

Casti does deserve a medal for even attempting to bring these stunningly
powerful ideas within reach of the rest of us. But the fact remains that the
cover blurb claiming that all who enjoyed the original will love this sequel is
the purest nonsense. The bulk of Five More Golden Rules will, I fear,
remain a closed book to anyone lacking a degree in mathematics.

It starts well enough, with an account of knot theory in general, and the
Golden Rule of the Alexander polynomial in particular. Casti’s potted account of
the triumphs and failures of knot theory, and its varied applications, is as
lucid as anything I’ve come across on the subject.

As I read on, however, I began to notice alarming changes in the pace of
Casti’s treatment. Like a passenger in a Ferrari with a dodgy turbo, one moment
I’d be pootling past a simple bit of algebra, and then—vrooom—I’d be
doing a ton and hurtling past a surface integral defined for closed curves in
R³.

It was the same with the next Golden Rule: the Hopf bifurcation theorem of
dynamical systems theory. Casti just gave me time to look at where we were
going—a trip around systems that evolve over time, for example ant
populations, and how the bifurcation theorem casts light on their stability, and
then—whoah—we’re off again, zooming past hyperbolic equilibrium
points as though they were traffic cones marking out a closed lane on a
motorway.

By the time we got to Golden Rule three, the Kalman filter of control theory,
I was wondering why the police—in the form of a decent editor—hadn’t
pulled Casti over. Again, had some idea where I was being driven—indeed,
control theory is all about whether you can get there from here—but I
could barely see what was racing by: pharmacokinetics, toy economies and trips
to the planets blurring and blending at high speed.

When we reached Golden Rule four, the Hahn-Banach theorem of functional
analysis, Casti put the pedal to the metal. I just wanted out. Thirty pages into
this chapter and even he was admitting he’d probably overdone the maths.

Don’t even think of asking me what the theorem means, just let me recite it:
“Every linear function defined on a subspace G of a vector space X can be
extended into the whole space X with preservation of norm.” So now you know.
Perhaps that’s just an elegant, mathematical way of describing how you plot a
function on a surface, then transform that surface into a 3-dimensional space,
like drawing a face on a balloon before blowing it up. Perhaps not.

Finally, we arrive, at Golden Rule Five, Shannon’s coding theorem in
communications. Casti’s eased off the gas. Great examples appeared of how the
theorem underpins everything from bar codes to DNA, and there was nothing harder
than logarithms to tax the reader. The theorem is a measure of the average
information in an event with its probability. Intuitively, we can understand
that if an event is highly unlikely, the knowledge that it has happened provides
us with a far more information, measured in bits, than knowing that something
that frequently occurs has happened.

Now I’m back on the road-side, I can see Casti’s supercharged jalopy has
taken me to some interesting places. Personally, I’m intrigued by how dynamical
systems act differently according to whether time is continuous or discrete.
What does this mean for the reliability of computer simulations, which always
use discrete time? And could it lead to tests of whether real time is quantised
or not?

I dare say there’s lots more food for thought in this book, if only I could
get my teeth into it more easily. Five More Golden Rules could have been
at least as successful as its predecessor—if only someone had put in the
time and effort to turn it into a coherent whole. That might have spared the
reader such absurdities as being confronted by a fearsome double integral early
on in the book and having to wait for more than a hundred pages to be informed
that the total area under the graph of a function is something called an
integral.

Despite my great disappointment with this book, I’ll still be dipping into it
in the hope that one day I might get used to Casti’s lurching driving.
Certainly, anyone who sees themselves as a mathematical Schumacher should
consider accepting Casti’s offer of a ride. L-drivers and pedestrians, however,
are best advised to wave him on.