IN 1994, John Nash won the Nobel Prize for Economics for a dissertation that
was 44 years old.

Which was nothing short of a miracle given the roller coaster life he had led
between 1950 and 1994. This brilliant mathematician oscillated between moments
of great genius and long periods of terrible mental illness, sometimes so severe
that he was confined in a mental institution.

So what was so significant about those 27 pages written at Princeton
University when Nash was just 21 years of age? Nothing less than a breakthrough
in game theory, the field pioneered in the 1920s by one of the fathers of the
modern computer, John von Neumann, in an attempt to use mathematics to analyse
human conflicts and predict their likely outcomes.

During the Second World War, the US Navy made heavy use of game theory in
antisubmarine warfare. So when the book The Theory of Games and Economic
Behavior was published in 1944, written by von Neumann and his colleague
Oskar Morgenstern, there was considerable military funding available to support
mathematicians, political scientists and economists drawn to the new field. In
the late 1940s game theory was hot.

Nash, a tall, well-built and strikingly good-looking young man with
mathematical talent and a fiercely competitive nature, was one of the first to
be attracted to the subject. One of the glaring holes in the theory as expounded
by von Neumann and Morgenstern was that it applied only to games involving two
combatants, or games that, through the formation of coalitions, could be
considered as two-party. The central mathematical theorem underpinning the
entire theory, von Neumann’s “minimax theorem”, could be proved only under these
stringent conditions. But for game theory to be really useful, it would have to
cover games with three or more players, each fighting to win on their own.
Hungry for a major result to make his reputation, he immediately looked for a
way to extend the theory to cover multiparty games.

Nash could never be bothered to absorb the existing literature in the field.
Instead, he preferred to begin at the beginning and work everything out for
himself. So whenever he approached a new problem, he did so with a fresh pair of
eyes.

His work on game theory demonstrated the originality he was to display in
later work. He proved that every game reaches a state of equilibrium where none
of the players can improve their position. This “Nash equilibrium” was the
mathematical basis for the nuclear standoff adopted by the major world powers
from the 1950s onwards.

With this result under his belt, Nash soon found himself working at the
government-supported RAND think-tank in Santa Monica, California. He was not
satisfied. While his work on game theory was highly regarded by military
strategists and economists, most mathematicians were not impressed. The strength
of game theory lay in its applications and the maths itself was fairly
straightforward. Nash wanted to be a great mathematician at a major university,
so even eminence in game theory wouldn’t do it.

He did show that he was a great mathematician by going on to prove major
results in two branches of mathematics. And by the time he was 29, Nash was a
tenured professor at MIT, widely regarded as one of the greatest mathematicians
of our time. The world was at his feet.

And then, with frightening rapidity, everything fell apart as he succumbed to
what was to be diagnosed as paranoid schizophrenia. He suffered wild delusions:
he saw secret messages coded for him alone in newspaper headlines, God was
calling him to form a world government. Needless to say, he lost his job.

During the years that followed, he would have brief periods when the illness
receded and he produced work that most mathematicians would be proud of, even
though it didn’t show the brilliance of his early work. But to those of us who
follow his life through Sylvia Nasar’s biography, the bulk of his adult life
looks like sheer hell.

I found Nasar’s book a “must read”, with something for everyone. The first
third provides a good overview of the rise of American mathematics during the
late 1940s and through the 1950s, particularly the role played by Princeton.
Spurred on by the Cold War and the space race, these were the golden years of
modern American mathematics, when funds flowed freely and American
mathematicians—or, rather, mathematicians who went to live
there—began to lay the foundation of their present international
dominance. Nash is just one of several characters we encounter. To her credit,
Nasar, an economics correspondent for The New York Times, does not
underestimate her readership by leaving the mathematics out, though of necessity
she does little more than state some of the deep problems Nash and others
tackled.

Then the tenor of the book changes, as it leads us through Nash’s mental
journey. It’s a journey that reminds us of the price we may have to pay for our
creative, problem-solving minds. To achieve greatness in mathematics, you have
to create an entire world in your mind, and live in it almost exclusively for
months if not years on end. You have to have a strong compulsive streak that
continues to drive you forward when most would give up. The majority of the many
thousands of mathematicians in the world are lucky: their friends and colleagues
might merely label them as somewhat obsessive, occasionally distracted.

In Nash’s case, however, the line separating genius and madness seemed
woefully thin, easily crossed. The first major breach took place in 1959. After
that, the brief periods when Nash emerged from his private world and engaged in
mathematics were separated by years when the world he lived in was clearly not
ours.

But Nasar does not simply describe. By giving us details of Nash’s childhood
and his personal life, she tries to explain the other factors that led to, or at
least foretold, his eventual breakdown.

Finally we reach Nash’s later life, when the once-handsome young man has
become an unkempt, stooped, frail and cadaverous-looking old man, hanging around
the Princeton campus, writing crazy messages on blackboards. This was the period
when Nash was the ghostlike individual known as “The Phantom”, the once
brilliant mathematician who had “gone mad”. And yet, remarkably—and
confounding the accepted medical wisdom concerning schizophrenia, that it rarely
recedes—during the Phantom years, the wildfire in Nash’s mind began to die
down. Life became “normal”. He even found himself able to do mathematics
again.

And then the dramatic ending: the awarding of the Nobel prize. Here, Nasar
excels herself as a journalist, somehow managing to break through the wall of
secrecy that normally surrounds the deliberations of the Nobel Prize Committee
to tell us the inside story of the bitter struggle, and the narrow vote, that
led to Nash being chosen. Suddenly, Nash, the Princeton Phantom, found himself
treading the world stage. He had finally achieved the fame and recognition he
craved as a young man.

Today, Nash lives a quiet life in Princeton, together with his former wife,
and in touch with his two sons, one of whom also suffers from schizophrenia.
Given all that went before, it’s probably as happy an ending as there can be. In
many ways, Nash’s life has been a remarkable mental journey. Nasar’s first-rate
biography describes that journey for us. But I doubt any of us would want to
travel it ourselves.

NOT long after he became the first man to walk on the Moon, astronaut Neil
Armstrong visited Britain and was guest of honour at a celebration at 10 Downing
Street. During the course of the evening, he proposed a toast to the man who, he
said, had played the most significant role in the history of crewed space
flight. Who was that man?

Aristotle? Galileo? Isaac Newton? Wernher von Braun? John F. Kennedy? My full
list would have been much longer, but the name Armstrong toasted was not on it.
Chances are it wouldn’t be on your list either. Who exactly is, or was, John
Harrison, the individual who according to Armstrong played such a pivotal role
in the exploration of space.

Harrison, who lived from 1693 to 1776, was a carpenter and self-taught
clockmaker who built the world’s first truly reliable, accurate timepiece. (His
story is told at length by Dava Sobel in Longitude.) Given NASA’s
significant ambassadorial role, it seems likely that part of the motivation for
Armstrong’s choice was that Harrison was a Yorkshireman, as was Britain’s prime
minister at the time, Harold Wilson. But international diplomacy aside,
Armstrong still had a point.

If you can’t yet quite see what that point is, then you have obviously not
read Jo Ellen Barnett’s fascinating new book Time’s Pendulum, a
300-plus-page account of humanity’s long struggle to measure time. And you
should.

As Barnett makes clear, a great deal of the early part of that effort
amounted to trying to decide what exactly it was that should be measured. The
basic unit of time was the day—from sunrise to sunset—as it had been
since sundials were invented in ancient Egypt around 1500 BC. The Egyptians were
the first to break the day into 12 equal parts, giving us the forerunner of
today’s hours. As Barnett points out, the length of the Egyptian hour was not
fixed: it varied with the length of the day, and hence with the seasons.
Moreover, for the ancient Egyptians, time as something measurable ceased during
the hours of darkness.

Without a sunny climate, something more than a sundial was needed. The desire
for a way to measure time independently of the Sun gave rise to myriad devices,
most notably sandglasses, water clocks and candles. All three provided a
metaphor for time as something flowing continuously, and began to shape the way
we think of it.

Though their accuracy was never great, sandglasses, water clocks and candles
not only provided a way to measure time when the Sun was not visible in the sky,
they also provided the basis for a concept of time that did not depend on the
length of the day. But it was to be many centuries before anyone took advantage
of that possibility. As recently as the 13th century, time was still conceived
as a division of the daylight period. The various devices for measuring it came
with elaborate procedures to adjust them to measure the ever changing hours.

It was into that world of “natural time”, based on the Sun’s march across the
sky and varying with the seasons, that the first mechanical timepieces were
introduced in 13th-century Europe. At odds with the conception of time as
something that flows, with the introduction of the first clocks came the idea of
measuring time by splitting it into discrete chunks and counting them.

That the early clocks were highly unreliable was of little consequence
because they could be checked and adjusted regularly by reference to the Sun. So
despite the technology, time still depended on the Sun, and still varied from
season to season. The “time” given by a mechanical device was not considered to
be the real time and had to be indicated as such, by means of the phrase “of the
clock”, later abbreviated to “o’clock”.

Underlying the development of ever more accurate clocks came a new conception
of time as something that flows—or ticks—of its own accord, in a
uniform fashion, independent of the rotation of the Earth or its motion around
the Sun. As Barnett points out, this view of time has become so ingrained that
it is hard to step back and realise that time is a human invention, something
that exists, in a practical sense, only by virtue of the machines we develop to
“measure” it. (In point of fact, the thing we are measuring is created by the
devices that do the measurement.)

The introduction of accurate clocks also enabled sailors to make use of the
variation of time with longitude to determine their longitudinal position at
sea. This motivated Harrison to perfect his timepiece, and gave Armstrong a
conversational gambit two centuries later.

Another development in the ever changing concept of time was brought about by
the growth of the railways in the 19th century, particularly in North America.
With reliable clocks, it was possible for people within towns to synchronise
their daily activities. Rail travel necessitated coordination of all those
different local times. The end result of this change is our system of time
zones, with a uniform notion of time within each zone. After two thousand years,
a completely abstract, human-made notion of time had been put in place. Human
life would never be the same.

And now, although I have more time, I find I am out of space. But there is
still three-quarters of Barnett’s story to go. After taking us to today’s
definitive caesium clock, accurate to one second in three million years,
Time’s Pendulum embarks on a second journey, following a second human quest
for time: how science has found ways to measure the age of the Earth and put a
timeline on the species that have inhabited our planet.

Much space on today’s popular science shelves is taken up with accounts of
the nature of the Universe or the mystery of consciousness. With Time’s
Pendulum, Barnett has shown us that there is a mystery and a great story to
be found in the very time that flows past us and which rules our lives. There is
no need for wild speculation. Time’s Pendulum is history. In both
senses of an ambiguous phrase, it is the history of our time.

THE deadline was approaching. It was Thursday evening, and New
ĐÓ°ÉÔ­´´ wanted my review by first thing Monday morning. I had an
inch-thick volume to get through. No matter, I had done this many times before
and can get through most popular science books in about three hours. All was
going smoothly until I read this:

“I write a lot of book reviews. It’s a great way to do three things at once:
bone up on a new area, acquire a free book, and get paid for doing the above.
It’s also excellent practice in speed reading (the magazine or newspaper always
wants the review by yesterday) and quick-fire journalism (ditto). New
ĐÓ°ÉÔ­´´ had signed me up to review a book called . . .”

I was caught in the act. And in the process, became an unsuspecting
participant in another spin that can be put on that age-old favourite of popular
books on mathematics—the self-referential paradox, such as the person who
gets up and says “I am lying”.

Now, I am sure that 90 per cent of the readers of this review do not need me
to go on and ask the question, “Was the person lying or telling the truth when
he or she made that assertion?” (If you are one of the other 10 per cent, now’s
your chance to figure it out.)

It would be easy to dismiss The Magical Maze as yet another rehash
of the familiar fare of popularisations of mathematics—including
self-referential paradoxes. But that would be unfair for at least two reasons.
First, Stewart is one of an extremely small group (in mathematical parlance,
“vanishingly small”—that’s an insider’s joke for mathematicians) who have
helped shape the form and content of the modern genre of mathematics
popularisation. Second, in some ways it is that 10 per cent of new readers that
mathematics popularisers must constantly strive to reach to ensure that the
supply of new mathematicians does not dry up. And third, one of the creative
aspects of writing about mathematics for a general audience is to keep finding
new ways to present the same material, to take the old and familiar and make it
fresh and relevant to a new generation.

Wait a minute. That’s three reasons. Didn’t I say there are at least two?
Let’s look back to the start of that last paragraph. Yup! I said “at least two”
all right. So I was wrong, right? Well, no. “At least two” includes there being
exactly three, you see. That’s how mathematicians use language.

Come to think of it, I might as well give a fourth reason—four is also
“at least two”. Many people like to take a new journey through familiar
territory, an observation that applies to popularisations of mathematics every
bit as much as to singers, orchestras and movie producers.

If you have enjoyed reading the review so far, you will undoubtedly like
Stewart’s own chatty, jokey style, which I have just tried to adopt. If you
prefer your popularisations of mathematics to be presented in a more sombre
style, then the later Stewart is not for you (though the earlier Stewart is).
This one is based on his Michael Faraday Christmas Lectures for Young People,
broadcast by the BBC. Avid readers of popular accounts of mathematics will find
little new in terms of content. Nor, for that matter, will avid readers of
Stewart’s other popular mathematics books.

Topics include Fibonacci numbers, the Monty Hall problem, slime moulds and
emergent behaviour, symmetry, chaos, fractals, Turing machines, minimal
surfaces, Gödel’s theorem—and self-referential paradoxes. All
familiar fare, but, as I have tried to explain, that is not the point.

Jan Gullberg, a Swedish surgeon, has offered up a huge dollop of nostalgia in his 1100-page volume, Mathematics, From the Birth of Numbers. For any British scientist educated in the 1950s or 1960s, the book will surely bring back many childhood memories.

The first memory will be of those heavy volumes, found in even the most non-bookish household, that promised to tell you everything about the human body or the world of science, or even claimed to offer up “everything about everything”. In part encyclopedias, these books showed that their authors had laboured long and hard to develop themes and narrative lines—in short, readability. As a result, no matter where you delved, you would find yourself being drawn into the thread. They were a boon for any harassed mother faced with an inquisitive teenager kept home sick in bed, unable to go to school.

The second memory will be of what, in the Britain of the two decades just after the Second World War, was called A and S-level mathematics. And this is almost exactly the point at which Gullberg’s volume ends. After a long introductory section on numbers and number systems, the bulk of the material covered either focuses on, or is centred around, calculus. Looking back to my own school days, this is almost exactly the view I had of mathematics at the time.

Thirty years later, having spent the entire intervening period as a professional mathematician, I realise just how narrow and, as a result, uncharacteristic that view was, even for the mathematics of the time. But back then it excited me and was enough to persuade me that mathematics was the life for me.

Mathematics is one of the great creations of the human mind, a magnificent part of our culture, and it bothers me to see it maligned—as it so often is these days—almost always as a result of ignorance. I am also bothered when the true nature of the subject is misrepresented.

Gullberg is quite clearly a mathophile, a nonmathematician who loves mathematics, so there is no way he is going to malign the discipline. But he does misrepresent it—not intentionally, but in exactly the same way I would have had I sat down and tried to write a book about mathematics when I was fresh from my GCE A-level and S-level studies. He gives us a lot of breadth in essentially one part of mathematics, but little by way of depth. Large parts of the discipline are not even mentioned.

So, as a “general treatment of the field of mathematics”, which seems to be how the book is being marketed, I think it fails. But when I put the marketing to one side, what remains is a fascinating, though curious, addition to the mathematical literature. Gullberg has developed an interest in—even a love of—mathematics, that has inspired him to spend 10 years of his life learning about the subject and putting what he has learnt into this book.

And that interest shines through. Not on every page, since large sections of the book read like lecture notes that have been cleaned up and filled out for publication. Rather, it is when you look at the work as a whole, at the care and attention that has gone into its preparation, that you cannot but admire the author’s dedication and commitment.

Yet what Gullberg has given us is not the insider’s view of mathematics. How could he? He is not a mathematician. Instead it is very much the impression of a keen amateur, from the outside. And while those of us “in the business” may find fault with the result, about 10 000 readers in the US have already decided that what he has to say is sufficiently interesting to buy the book.

This begs the really fascinating question of why so many people are buying it. Certainly not because it describes any new mathematics. Indeed, the view of mathematics it gives is for the most part frozen in the 1950s. Even the occasional mention of more recent developments, such as the very brief treatment of fractals, has a dated look to it.

The reason for the book’s success, I would suggest, is bound up with the notion of nostalgia that I began with. Mathematics has changed so much during the past 200 years that much of what you see in even the best popular expositions of mathematics is alien territory for most people. It does not resemble the mathematics that they learnt as teenagers. (The same can be said for physics and, indeed, any of the sciences.)

To see what I mean, take a look at any of the more recent popular books on mathematics: William Dunham’s The Mathematical Universe, Ian Stewart’s two books Nature’s NumbersandThe Problems of Mathematics, John Casti’s Five Golden Rules, or my own Mathematics: The Science of Patterns. Those books try to convey some of the incredible variety of modern mathematics. But in order to do so in a way accessible to the general reader, they all have to leave out most of the detail. In that respect, they may well leave some readers unsatisfied in a way that Gullberg’s book does not.

For many readers now in their late forties and upward, who have not studied mathematics since adolescence, Gullberg’s book provides a reassuringly familiar look at a field that in their youth they found fascinating. They, I suspect, are the ones buying the book. And since no mathematician would dream of writing such a book, Gullberg has a clear field.

So I am left in two minds. On the one hand, I am bothered that Gullberg’s book presents such a narrow, dated view of mathematics. On the other hand, Gullberg’s book does provide a well-crafted encapsulation of a view of mathematics that is not uncommon among mathematically literate nonmathematicians of a certain age.

If I had to vote on my dilemma, I would come down on Gullberg’s side, and for this reason. Let’s suppose I am right, and that middle-aged people are buying the book because it offers them the promise of being able to really “get back into” some serious mathematics that looks familiar to them. Mathematics, From the Birth of Numbers then becomes probably the only book with genuine mathematical content they possess. Then, the next time a sick son or daughter—or maybe a grandson or granddaughter—has to miss school and stay home in bed because of illness, Gullberg’s book is brought out to while away the hours. By the end of that week in bed, maybe—just maybe—a spark of interest will have been kindled in a young mind, and a future mathematician will have been born.

The fact is, if I had been such a sick teenager, Gullberg’s book would have had exactly that effect on me. In the final analysis, that alone is reason enough for me to say thank you to Gullberg for writing it.

THE story so far: episode 1 takes place in 1989, when the brilliant and
charismatic Oxford mathematician and physicist Roger Penrose writes a book
called The Emperor’s New Mind. The book attacks artificial intelligence
(AI) as a theory of mind and proposes a radically new theory based on
practically all of modern physics. Although heavy going, it swiftly becomes a
bestseller around the world.

Mathematicians, philosophers, cognitive scientists, physicists and countless
others around the world mount separate, but sustained attacks on both Penrose’s
overall programme and on specific parts of his argument. The AI folk were up in
arms as well, but as a certain Mandy Rice Davis said long ago in quite a
different context, “Well, they would say that, wouldn’t they?”

Episode 2 occurs in 1994 when, responding to his critics, our hero writes
Shadows of the Mind. Fully half of this is an attempt to refute
technical objections that had been levelled against The Emperor’s New
Mind.

In Shadows of the Mind, Penrose tries to clarify what he sees as the
physicist’s or mathematician’s fundamental view of existence, in particular the
relationship between the physical world, the mental world and the Platonic world
of ideas in which mathematicians and physicists work.

But to no avail. The criticisms continue unabated. Our hero decides to write
another book. This time he will invite critics to comment on his ideas and then
respond to their points, and the whole package will be published together. Now
read on…

The Large, the Small and the Human Mind is the third episode in the
saga, and is considerably shorter than the previous two. The large of the title
is the physics of the Universe. The small is quantum physics. The human mind is
. . . well, that is precisely the question Penrose wants to answer. What exactly
is the human mind? What gives rise to intelligent behaviour and to our perceived
free will and sense of consciousness? He believes the answers to questions about
the mind can be found in a future marriage between the physics of the large and
of the small.

It is a bold suggestion, perhaps just wild enough to be near the truth. Then
again, I haven’t sufficient familiarity with big-bang cosmology or quantum
physics to follow all Penrose’s arguments in detail, let alone fling various
notions around with the seeming abandon of a master such as Penrose, so I cannot
confidently judge how likely he is to be correct. And I assume the same is true
of most of his readers.

But that doesn’t matter. What all three books present is science in the
making. For the most part, the stuff that makes its way into science books is
the stuff that survives. The mistaken ideas are simply forgotten. The result is
that nonscientists are presented with a highly misleading view of science and of
the way scientists work. To see a scientist of Penrose’s ability, stature and
achievement toss large parts of modern physics into the air as though juggling
balls and try to keep them aloft while marshalling them into a coherent pattern
is a thing to behold. It is a wonderful illustration of a first-rate scientist
doing what first-rate scientists have always done: make bold conjectures and
display them for others to confirm, refute or amend.

Of course, from a scientific point of view, the fundamental question is
whether or not the argument Penrose presents turns out to be right. But I
believe the real value of his books, written so that a general audience can
follow the gist of the argument, if not the detail, lies in the fact that he is
doing in public what most scientists generally do in private.