Ian Stewart is an Emeritus Professor of
Mathematics
at Warwick University and a Fellow of the Royal Society. He has written
over 80 books, mainly popular mathematics, and has won three gold
medals for his work on the public understanding of science. In
collaboration with Terry Pratchett and Jack Cohen he wrote the
Science of Discworld series. His new book, 17 Equations That Changed the World, is published by Profile.
Buy 17 Equations That Changed the World from the Guardian bookshop
"'Popular
mathematics' may sound like a contradiction in terms. That's what makes
the genre so important: we have to change that perception. Mathematics
is the Cinderella science: undervalued, underestimated, and
misunderstood. Yet it has been one of the main driving forces behind
human society for at least three millennia, it powers all of today's
technology, and it underpins almost every aspect of our daily lives.
"It's
not really surprising that few outside the subject appreciate it,
though. School mathematics is so focused on getting the right answer and
passing the exam that there is seldom an opportunity to find out what
it's all for. The hard core of real mathematics is extremely difficult,
and it takes six or seven years to train a research mathematician after
they leave school. Popular mathematics provides an entry route for
non-specialists. It allows them to appreciate where mathematics came
from, who created it, what it's good for, and where it's going, without
getting tangled up in the technicalities. It's like listening to music
instead of composing it.
"There are many ways to make real
mathematics accessible. Its history reveals the subject as a human
activity and gives a feel for the broad flow of ideas over the
centuries. Biographies of great mathematicians tell us what it's like to
work at the frontiers of human knowledge. The great problems, the ones
that hit the news media when they are finally solved after centuries of
effort, are always fascinating. So are the unsolved ones and the latest
hot research areas. The myriad applications of mathematics, from
medicine to the iPad, are an almost inexhaustible source of
inspiration."
The
self-taught Indian genius Srinivasa Ramanujan had a flair for strange
and beautiful formulas, so unusual that mathematicians are still coming
to grips with their true meaning. He was born into a poor Brahmin family
in 1887 and was pursuing original research in his teens. In 1912, he
was brought to work at Cambridge. He died of malnutrition and other
unknown causes in 1920, leaving a rich legacy that is still not fully
understood. There has never been another mathematical life story like
it: absolutely riveting.
One
of the great cult books, a very original take on the logical paradoxes
associated with self-reference, such as "this statement is false".
Hofstadter combines the mathematical logic of Kurt Gödel, who proved
that some questions in arithmetic can never be answered, with the
etchings of Maurits Escher and the music of Bach. Frequent dramatic
dialogues between Lewis Carroll's characters Achilles and the Tortoise
motivate key topics in a highly original manner, along with their friend
Crab who invents the tortoise-chomping record player. DNA and computers
get extensive treatment too.
In
his long-running Mathematical Games column in Scientific American,
Gardner – a journalist with no mathematical training – created the field
of recreational mathematics. On the surface his columns were about
puzzles and games, but they all concealed mathematical principles, some
simple, some surprisingly deep. He combined a playful and clear approach
to his subject with a well-developed taste for what was mathematically
significant. The book consists of numerous selections from his columns,
classified according to the mathematical area involved. Learn how to
make a hexaflexagon and why playing Brussels sprouts is a waste of time.
4. Euclid in the Rainforest by Joseph Mazur
A
thoroughly readable account of the meaning of truth in mathematics,
presented through a series of quirky adventures in the Greek Islands,
the jungles around the Orinoco River, and elsewhere. Examines tricky
concepts like infinity, topology, and probability through tall tales and
anecdotes. Three different kinds of truth are examined: formal
classical logic, the role of the infinite, and inference by plausible
reasoning. The story of the student who believed nothing except his
calculator is an object lesson for everyone who thinks mathematics is
just 'sums'.
In
1852 Francis Guthrie, a young South African mathematician, was
attempting to colour the counties in a map of England. Guthrie
discovered that he needed only four different colours to ensure that any
two adjacent counties had different colours. After some experimentation
he convinced himself that the same goes for any map whatsoever. This is
the remarkable story of how mathematicians eventually proved he was
right, but only with the aid of computers, bringing into question the
meaning of "proof". It contains enough detail to be satisfying, but
remains accessible and informative throughout.
The
classic text What is Mathematics? by Richard Courant and Herbert
Robbins focused on the subject's nuts and bolts. It answered its title
question by example. Hersh takes a more philosophical view, based on his
experience as a professional mathematician. The common working
philosophy of most mathematicians is a kind of vague Platonism:
mathematical concepts have some sort of independent existence in some
ideal world. Although this is what it feels like to insiders, Hersh
argues that mathematics is a collective human construct – like money or
the Supreme Court. However, it is a construct constrained by its own
internal logic; it's not arbitrary. You choose the concepts that
interest you, but you don't get to choose how they behave.
Both
authors are top-rank mathematicians with years of stage performances
behind them, and their speciality is mathematical magic. They show how
mathematics relates to juggling and reveal the secrets behind some
amazing card tricks. Here's one. The magician mails a pack of cards to
anyone, asking them to shuffle it and choose a card. Then he shuffles
the cards again, and mails half of them to the magician—not saying
whether the chosen card is included. By return mail, the magician names
the selected card. No trickery: it all depends on the mathematics of
shuffles.
Biologists'
understanding of many vital features of the living world, such as sex
and survival, depends on the theory of evolution. One of the basic
theoretical tools here is the mathematics of game theory, in which
several players compete by choosing from a list of possible strategies.
The children's game of rock-paper-scissors is a good example. The book
illuminates such questions as how genes spread through a population and
the evolution of cooperation, by finding the best strategies for games
such as cat and mouse, the battle of the sexes, and the prisoner's
dilemma. On the borderline between popular science and an academic text,
but eminently readable without specialist knowledge.
A
collection of 23 science fiction short stories, each of which centres
on mathematics. Two are by Martin Gardner, and many of the great writers
of SF are represented: Isaac Asimov, Gregory Benford, Larry Niven,
Frederik Pohl. The high point is Norman Kagan's utterly hilarious "The
Mathenauts", in which only mathematicians can travel through space,
because space is mathematical – and, conversely, anything mathematical
can be reality. An isomorphomechanism is essential equipment. Between
them, these tales cover most of the undergraduate mathematics syllabus,
though not in examinable form.
There
ought to be a great classic in this top 10, and there is none greater.
I've put it last because it's not popularisation in the strict sense.
However, it slips in because it communicated to the world one of the
very greatest ideas of all time: Nature has laws, and they can be
expressed in the language of mathematics. Using nothing more complicated
than Euclid's geometry, Newton developed his laws of motion and
gravity, applying them to the motion of the planets and strange wobbles
in the position of the Moon. He famously said that he "stood on the
shoulders of giants", and so he did, but this book set the scientific
world alight. As John Maynard Keyes wrote, Newton was a transitional
figure of immense stature: "the last of the magicians … the last
wonderchild to whom the Magi could do sincere and appropriate homage."
No mathematical book has had more impact.
