The Universe and the Teacup by K. C. Cole, Little, Brown, 拢12.99, ISBN
0316644250
A YEAR or so ago, I decided to spend the winter evenings learning to paint
with watercolours. I signed up for a course, and the teacher told us we would
spend a little time learning the basic techniques鈥攕tretching paper, laying
down a wash鈥攖hen would quickly get down to painting. As the lessons
progressed, he said, we would not just become proficient with watercolours: we
would also start to 鈥渟ee鈥� the world around us for the first time鈥攖he true
colours and shapes of trees and hills and clouds.
Well, he was half right. I proved to be very much at the car-respray end of
the artistic spectrum, but having spent hours struggling to capture the form of
everyday things with a few brush strokes, I certainly found myself far more
visually aware. I once caught myself staring at some motorway scrubland trying
to work out what mix of paints I鈥檇 need to get it down on paper.
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What struck me about the course was the swiftness with which it progressed
from tools of the trade to real creation (and many of my classmates proved to be
talented watercolourists).
The contrast with the way mathematics and science are taught at school could
hardly have been greater. Lesson after lesson, term after term, year after year:
ever more tools, ever more technique鈥攁nd nothing more creative than exam
questions to practise on. I suspect we would not tolerate art schools in which
students learnt nothing but how to mix paint. Yet is our approach to science and
mathematics teaching much more than ever more complex paint-mixing?
I therefore have a modest proposal. Scrap all mathematics and science
syllabuses forthwith, and replace them by the equivalent of that watercolour
class. Abandon the stultifying and alienating emphasis on ever-more-pointless
technique, and stress instead ways of 鈥渟eeing鈥� the science and mathematics in
the world around us.
Fortunately, we don鈥檛 have to wait for some costive slug in Whitehall to give
us the green light for this modest proposal. For in The Universe and the
Teacup, K. C. Cole has written a nice set text: a first course in
mathematics appreciation.
Most mathematicians would probably find it hard to step back from their
subject and give wide-ranging examples that convey the fascination of
mathematics to a lay audience. But Cole isn鈥檛 a mathematician鈥攕he is a
journalist with the Los Angeles Times, which makes her well placed to
spot the most exciting aspects of modern mathematics. She begins with two
concepts that often catch us out: exponential growth and risk.
With a well-chosen anecdote typical of the rest of the book, Cole retells the
legend of how the inventor of chess was asked by a king what she (for it is she,
according to Cole) would like as a reward. She asks merely for two grains of
wheat on the first square of a chessboard, four on the second and so on. The
total winnings are likely to surprise most people: 2 to the power 64 doesn鈥檛
sound like a whole lot, but by the time exponential growth has worked its magic,
it amounts to about 10 million million tonnes of grain.
Cole then goes on to show how a vast number of natural phenomena exhibit this
mathematical effect, from the birth of humans to the birth of the Universe, and
how failing to spot it can lead us astray. As she points out, a population
growth rate of 1.9 per cent per annum sounds like nothing to worry
about鈥攗ntil one works out that exponential growth will double a population
in just 36 years.
Cole deals with risk with equal facility, opening our eyes to the traps that
await us if we rely only on our common sense. Indeed, much of her book is less
about showing how detailed mathematical techniques can put us straight than
about showing how simply being aware of them can alert us to dangers ahead. For
example, in the second part of her book鈥斺�滻nterpreting the Physical
World鈥濃�攕he shows how knowing even a little about nonlinear phenomena in
the natural world is enough to make one sceptical of claims that it is possible
to predict such things as climate change with any confidence.
Having a personal interest in using mathematics in fields generally thought
beyond analysis, I found the third section鈥� 鈥淚nterpreting the Social
World鈥濃�攖he most intriguing of all. Cole points out that even a cursory
knowledge of Ken Arrow鈥檚 Impossibility Theorem exposes the fatuity of
all attempts to find the 鈥減erfect鈥� voting system, before showing how Fair
Division theory makes possible the seemingly unattainable: a fair, envy-free way
of dividing up an inheritance.
The final section鈥斺�漈he Mathematics of Truth鈥濃�攚as, in truth, a
little disappointing. Cole deals with the use of probability to establish proof
in both science and law, yet fails to mention the one key mathematical result
underpinning it all: Bayes鈥檚 Theorem. Still, the book ends well, with a chapter
on symmetry, broken symmetry and invariance. This encompasses everything from
the Thalidomide tragedy鈥攗ltimately tied to the asymmetry of the molecules
involved鈥攖o Emmy Noether鈥檚 theorem linking symmetry to conservation laws
and its use in the search for a Theory of Everything.
The book does have its faults. In some chapters Cole seems to lose confidence
in her grasp of the subject and relies on just too many anecdotes and quotes
from other popularisations to make her case; for example, the signal-to-noise
ratio in the chapter on noise will be far too low for many readers of New
杏吧原创.
Some basic numerical assertions are wrong: the chances of pulling two aces
from a deck of cards are much higher than 1 in 400, 10 to the power 26 is not 10
with 26 noughts after it, and the Sun emits a sight more than 鈥渟everal ocean
liners worth of mass each day in the form of radiant energy鈥濃�� a few
million times more, in fact.
Still, these points鈥攚orthy of Dickens鈥檚 schoolmaster Mr
Gradgrind鈥攕hould not deter anyone from buying this book: as a
wide-reaching and accessible introduction to what mathematics can do for us all,
it has few peers. After reading it, I predict you鈥檒l find yourself 鈥渟eeing鈥�
little nuggets of mathematics shining out at you pretty much everywhere.
And if you say your school maths lessons made you see quadratic equations
everywhere, I鈥檓 afraid I just don鈥檛 believe you.