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Twenty-three centuries ago Euclid compiled the most influential book
in all mathematics, The Elements. The elegance with which he proved the
key discoveries of Greek geometry has entranced mathematicians ever since.
He also established a simple paradigm: mathematics is what you can deduce
through a series of logical steps. On the rock of that paradigm, mathematicians
built the language of modern science. Now, 2000 years on, the first cracks
in the paradigm are beginning to show. And they are fracturing the world
of mathematics.

The cause of this disturbing turn of events is the computer. Invented
by one generation of mathematicians and dismissed as a toy by the next,
this handy algorithm processor has come back to haunt today’s generation.
By giving mathematicians the ability to do billions of complicated calculations
on their own desks, the computer has spawned a whole new way of doing mathematics
known as experimental maths.

Instead of deducing proofs step by step, experimental mathematicians
gain knowledge in the same inductive way as most other scientists. While
scientists design experiments on parts of the real world, the new mathematicians
experiment by looking for patterns in abstract worlds existing only inside
a computer. In the past, the result of a mathematical research project would
be a scientific paper carrying a proof or refutation of a theorem. Now it
can consist of some brightly coloured pictures and a joyous cry of ‘Look
what I found!’.

As for Euclid’s proofs, if they come into it at all they will probably
be someone else’s job. Some mathematicians are just too busy experimenting.

Although schoolchildren spend many years studying a subject called maths,
they see very little of what a mathematician does. Schools teach mainly
a series of computational techniques useful for things like counting your
change or calculating the trajectory of a cannonball. These techniques are
difficult to learn, which is why maths at school is hard. But professional
mathematicians are concerned with proving that the techniques work, which
is a quite different thing.

Deep down, all mathematics follows the example of Euclid, being founded
on a few axioms – the basic rules that define the area of study. Axioms
such as ‘the shortest distance between two points is a straight line’ are
assumed. Theorems such as ‘the sum of the angles in a triangle is 180 degrees’
are derived step by step from the axioms, using logical rules in a process
called a proof. Unlike other sciences, mathematics has remained a unified
subject in the 20th century. Indeed, the most spectacular research in recent
years has come from reuniting specialities. For example, combining calculus
with topology, a type of geometry where shapes can be stretched but not
cut, produced physicists’ favourite theory of the 1980s, superstrings.
But the emergence of experimental maths is causing the sort of schism that
has dogged much of the rest of science such as fundamental physics.

Mathematicians have traditionally spent their days dreaming up ways
of constructing a proof, armed only with a blackboard and some chalk in
a lonely struggle of mind against mystery. Now they may spend more time
playing with a personal computer than a 12-year-old with a fever for Tetris.
Some people have started calling themselves experimental mathematicians,
and they sit awkwardly with their traditional colleagues. There are departments
of experimental maths, and soon there will be a journal. Its editor is to
be David Epstein, a geometer at the University of Warwick in Coventry.

‘Originally, experimenting would have been doing calculations with a
pen and trying out various special cases of a theorem you think might be
true,’ says Epstein. ‘Then, when you’ve found enough cases to convince you
that it is true, try to prove it. This is a method (Karl) Gauss used a lot.
His private notebooks are just covered by huge numbers of calculations.’

In fact, such experiments are as old as maths itself. The Babylonians
discovered Pythagoras’s theorem about right-angled triangles hundreds of
years before Pythagoras was born. Later Pythagoras proved the theorem, but
it was discovered empirically. Practical people discovered it, it became
an accepted fact and then it was proved. This is how experimentalists would
like to see more maths progressing in future.

But for a long time, the culture that mathematicians work in has led
them to undervalue their experimental work. After all, the theorem is named
after Pythagoras, who proved it, not the Babylonian who discovered it. Epstein
himself has been guilty of hiding his experiments. ‘A typical example is
a 140-page paper I wrote and won a prize for,’ he says. ‘The whole thing
is based on computer work but the paper just goes on and on with theory.
The paper arose as a result of two years of investigation but only half
a page on the calculations went into the paper. The whole direction of the
research, how I decided which thing to try and do next, was determined experimentally
and half a page is all I gave it.’ Now Epstein wants experiments to come
out of the closet.

To discover Pythagoras’s theorem experimentally, the Babylonians must
have sat down and compared the size of the square on the hypotenuse with
the squares on the other two sides of many different triangles. Those calculations
could be done in the sand in a few days; to do the complex calculations
needed for today’s mathematics by hand takes much longer. The increasing
power of computers has made it easier to do these many calculations, which
has in turn encouraged experiments. Drawing a picture of chaos, for example,
only became possible with the advent of computers.

Although the computer is still an unfamiliar tool to many older British
researchers, most young American mathematicians are at home with it. There
are several programs, such as Steven Wolfram’s Mathematica, for doing the
routine calculations that used to take so long. Such programs take the drudge
out of solving many equations and have so made previously tedious areas
of mathematics attractive again. On top of this, a clutch of programs that
are designed for experimenting in special areas have appeared over the past
10 years. For example, the MacCauley program from Massachusetts Institute
of Technology, which has revitalised some parts of a generalisation of arithmetic
called ring theory, has been developed for more than five years. Axiom,
a new IBM program even proves some results.

For Epstein and his fellow experimenters, it is worthwhile taking the
time and trouble to write new software to do experiments because the experiments
nurture understanding and intuition about how mathematics works. Mathematical
experiments opened up new areas of study and new types of result, new techniques
and new ways of studying old fields. They feel that the creative spirit
of mathematics has been liberated. Other mathematicians, however, worry
that there may be a price to pay, too. Somewhere down the line, some mathematicians
fear they may be losing what makes maths special. Mathematics is so powerful,
they say, precisely because it consists of what is proven. In the past,
theories built on intuition have been flawed and theories based solely on
experiment probably will be too.

Widespread concern about too much experiment and too little proof has
led to a heated row over the quality of work done on fractals, the field
which to the outside world is the standard bearer of experimental maths.
Fractals, those wiggly shapes that, like Britain’s coastline, stay wiggly
however close you look, have enjoyed a recent boom in popularity. But they
have been around for 70 years. Gaston Julia, the French mathematician, was
proving theorems about them in the 1920s. Fractals remained largely unknown
because, unlike other shapes, fractals cannot be drawn with a pencil and
paper. ‘I thought it was my duty to study it, but it was not very interesting,’
remembers Lars Ahlfors, an 84-year-old geometer at Harvard, and winner of
the Fields Medal, the maths equivalent of the Nobel prize, who experienced
the dullness that buried the subject for a time.

Computers brought fractal geometry to life. In the 1960s Benoit Mandelbrot,
whose name is carried by the most famous of fractal sets, began to use computers
to draw fractals. Suddenly everyone could see the amazing things that Julia
had conjured up in his head. But it has turned out to be much easier to
draw a fractal with a computer than to prove a theorem about it.

For three years, Mandelbrot, and Steven Krantz, a mathematician at the
University of Washington at St Louis, have been locked in combat over the
worth of fractals. Krantz believes fractals, which have become so popular
that they are now mentioned in one-third of papers submitted to the leading
physics journal Physical Review Letters, are a mere fashion. There have
been a lot of pretty pictures but few theorems. He believes the subject
to be ‘easy, flashy, and as far as I can see, pointless’. Worse, some mathematicians
fear a Faustian bargain with experiment: pleasure now, pain later as they
clean up the messy proofs. Mandelbrot is unrepentant. Yes, proofs count
but work in fractals is imaginative and provocative. The study has deepened
our understanding of nature, and that is significant, whether or not you
believe it has solved any deep problems yet.

These competing assumptions of what maths is about pull in opposite
directions. Very crudely, Krantzians see mathematics as primarily a question
of proofs while Mandelbrotians see it as primarily a question of understanding.
Pursuers of understanding may leave proofs behind, chasers of proofs may
be missing chances to deepen their understanding. Proof or understanding,
Euclid or experiment: it is a battle for the soul of mathematics.

Epstein is trying to live with the contradiction. ‘My value system would
still value a proof higher. I agree with the other thing as well, that understanding
is the essential thing. But how do you demonstrate that you understand?
What would enrage most mathematicians would be if a person does a series
of computer experiments, says ‘Now I understand what is going on,’ and never
proves it. There is a temptation to do that.’

The tension between the methods of Euclid and experiment has come to
a head in a debate over the Mandelbrot set itself. One of the few big questions
in the field that has been solved is whether the Mandelbrot set is a collection
of isolated islands or a single connected whole. The Mandelbrot set looks
like a gingerbread man, but a closer look reveals mini-gingerbread men hovering
around the bulk of the set. The question is whether tendrils link the mini-gingerbread
men with the big one. The role of experiment in finding the answer is revealing.

As Krantz and others have pointed out, if you look at the pictures you
would think the set was unconnected (see graphic). However closely you look
at the set there are still isolated islands. But in fact, the truth is the
opposite: the Mandelbrot set is connected. The intuition that experi-ment
nurtures is false. The conclusion Krantz draws is that even in the heartland
of experimental maths, the new method can fail. Euclid 1, Experiment 0.

However, John Hubbard, of Cornell University, coauthor of the paper
that revealed the connectedness of the Mandelbrot set, remembers the discovery
differently. After staring at pictures of the Mandelbrot set for hours,
Hubbard became convinced that it was connected. The role of the pictures,
says Hubbard, was mainly inspirational. ‘It was more thought experiment
than computer experiment that led us to the conclusion the set was connected,’
says Hubbard. Nevertheless, he adds that after looking at the pictures:
‘We got a fixed belief it was connected on a local basis. We tried to imagine
what would happen under certain conditions and there seemed to be a topological
contradiction.’ He says that what he had was not quite a proof, but enough
understanding of what was going on to get one. Euclid 1, Experiment 1.

The unexpected connectedness of the Mandelbrot set is precisely why
mathematicians have for the past 100 years insisted on formal proofs. It
would have been easy to assume the Mandelbrot set was unconnected. In other
sciences that kind of assumption is made all the time. But, as the Victorians
discovered, in maths it can be lethal.

Monstrous mathematics

In the 19th century, a wave of apparent contradictions appeared in maths,
undermining the calculus of Newton and Gottfried Leibniz. In particular,
the Italian mathematician Guiseppe Peano discovered a monstrous line which
rebelled against its one-dimensional essence to fill up a two-dimensional
plane of space (see Figure above).

The problem was that mathematicians had got lazy. They had not followed
Euclid’s paradigm closely enough. Instead of strictly deriving their proofs
from axioms, they had been making assumptions and not proving theorems rigorously.
Their idea of a line was not precise, and because different assumptions
about a line could be built into different steps of an argument it was all
too easy to come up with contradictions. Many theorems were ‘proved’ by
demonstrating a single example and then generalising to all possible cases.

The remedy was to insist on a rigorous proof for every conjecture. Many
mathematicians, used to thinking in loose, intuitive ways, did not like
the idea. But the rigorous approach won out. As Hermann Weyl put it, logic
became the hygiene that mathematicians practised to keep their ideas healthy
and strong. And in mathematically polite society, you had to wash your hands
before you laid a theorem on the table.

When Victorian mathematicians grappled with the conflict between creativity
and rigour they chose, in the spirit of their times, rigour. Today, mathematics
is again grappling with the same conflict. But today our values are not
Victorian. The people pushing experimental mathematics hardest are young
Americans. A generation of scientists who grew up with free love now want
free mathematics. They will not be tied down by Euclid.

However, Bill Thurston says it is more a technological than a social
development. Computers have just become powerful enough to be useful to
mathematicians for the first time. ‘Mathematics is thinking, and what the
human brain can do is pretty amazing so the payoff point before the computer
helps is fairly high,’ he says. But Thurston has decidedly un-Victorian
motives, too. ‘It is a lot more fun, and it is changing all the time. People
are coming to grips with much more complicated phenomena.’ Experimenting
is for him a different way of tackling mathematical problems and he finds
it refreshingly free of established ways of doing things. It is nice, he
says, to see the young teaching the old.

Controversially, Thurston wants to see formal recognition for good experiments,
whether they have a theorem to go with them or not. He is unhappy with the
lack of recognition his experimental colleagues receive in some quarters
of the cloistered world of maths. In this vein, Epstein has recently proposed
institutional recognition of the shift away from proof towards understanding.
He wants doctorates to be awarded not just for theoretical work but for
writing useful computer programs, too.

The new generation of free-thinking mathematicians have learnt to use
the computer to change the pattern of their day-to-day work – and thinking.
The life of most mathematicians takes in months or even years of sterility
punctuated by bursts of intense creativity. Experimenting is one way of
bypassing the sterility because it instils belief in the truth of a theorem.
In maths it is often easier to prove a theorem once you become convinced
in your bones that it is true. If you are in doubt, trying to prove a theorem
true and then false, everything is much slower. Experiments can be like
stripes on the side of a car; they help you feel you are going faster.

Of course, if, as with the connectedness of the Mandelbrot set, you
are convinced of the wrong thing, you will be going in the wrong direction.
But plenty of discoveries have been made this way. The Victorians who discovered
the non-Euclidean geometries were trying to prove there was no such thing.
Progress in any direction is better than nothing. ‘I think a lot of people
get a lot more satisfaction from experimenting,’ says Epstein. ‘You have
all this abstract stuff, and even if you have trained as a pure mathematician
it can all get a bit much. To start relating it to numbers somehow makes
it feel less abstract. It certainly satisfies me.’

Mathematicians have always done experiments. They doodle, play and try
out examples all the time. It is part of the humdrum of mathematics. It
is the scale of the experimenting that is changing. Where in the past mathematicians
might have spent hours, even days, on a calculation, they might now spend
months or years in this phase.

Indeed, perhaps mathematics itself is entering an experimental phase.
After 23 centuries of living according to Euclid, perhaps mathematicians
are going to let their hair down a little. The Victorians thought they had
overthrown Euclid when they discovered non-Euclidean geometries. But they
kept on working in the same old ways. It may be only now, 100 years later,
that the promise of new freedom is finally to be fulfilled. Perhaps Euclid’s
Elements are not as elemental as we thought.