IT IS a number like no other. It is smaller than anything except zero, but
it鈥檚 not zero. It makes no logical sense, but it has endless uses. It鈥檚 the
infinitesimal, and it鈥檚 back.
Infinitesimally speaking, a circle is actually a polygon with infinitely many
infinitesimal sides. A solid is in fact an infinite sandwich of infinitesimally
thin slices. And velocity is an infinitesimal distance divided by an
infinitesimal time. The whole world is made of these next-to-nothings.
Yet 19th-century purists found these tiny slices of nothing just too much to
swallow, and they were banished from mathematics for more than a century. Only
recently has the concept been restored to respectability, and it鈥檚 back with a
vengeance. Infinitesimals are now giving us insights in physics and simplifying
our understanding of key areas of pure mathematics. But how on earth can such a
peculiar notion make sense?
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The idea of infinitesimals is an ancient one. Archimedes used them to
calculate the volume of a sphere. He imagined cutting the sphere into infinitely
many slices and hanging them on one arm of a balance. He rearranged them so that
they exactly balanced a cone hanging on the other arm. Knowing how levers work,
he related the volume of the sphere to that of the cone, and out popped his
formula. It鈥檚 a bizarre idea: if the slices are infinitely thin, how can they
have any weight or volume? How can you add up a lot of zeros to get something
substantial? Yet somehow it works.
The real arguments began after Newton devised his calculus, a way to
calculate rates of change, such as velocities. He looked at the changing
quantity (position, say) at two times separated by a very short interval, and
calculated a formula for the average rate of change in that interval. Then to
get the answer for a precise moment in time, rather than the average over a
short interval, he shrank the interval in the formula to zero. That is, he did
the calculation assuming it was non-zero, and then set it to zero. This
vanishing quantity he called a 鈥渇luxion鈥� and, like Archimedes鈥檚 slices, it
worked.
But again the logic seems faulty. Over an interval of zero seconds, the
position changes by zero, so the calculation becomes 0/0, and every
mathematician knows that 0/0 can be anything you like.
This was too much for the bishop and philosopher George Berkeley, who in 1734
published a pamphlet called The Analyst, or a Discourse Addressed to an
Infidel Mathematician. . . (the title went on a bit). He sarcastically
called Newton鈥檚 fluxions 鈥済hosts of departed quantities鈥�. His criticisms were
spot on, but people kept on using calculus because it always gave sensible
answers. It was more like magic than mathematics, but the spell worked.
Still, mathematicians in the 19th century were troubled. If a positive number
is not zero yet is still smaller than any positive non-zero number, then it must
be smaller than itself. That鈥檚 impossible. So they found alternative ways to set
up calculus, and called the resulting formalism 鈥渁nalysis鈥� to emphasise its
technically demanding foundations. Many people, especially school teachers and
students of calculus, looked wistfully over their shoulders, because
infinitesimals are much easier to handle than the rigours of analysis.
But mathematicians were determined: infinitesimals could not possibly exist.
That鈥檚 because the basic ingredient of analysis is the concept of a 鈥渞eal鈥�
number. And in the technical mathematical sense of the word real, a real number
is one that can be written as a decimal. Real numbers include all whole numbers
and all fractions, together with more subtle numbers like &pgr; and 鈭�2.
Crucially, no matter how small a real number is, there is always another number
smaller than it鈥攋ust divide 1 by some large enough number n. This
property of reals is called the Archimedean axiom, and it leaves infinitesimals
out in the cold, since it鈥檚 impossible to imagine a real number that is like 1
divided by infinity. It would be a decimal with infinitely many zeros鈥攊n
other words, zero itself.
It was more than a century later, in the 1960s, before the logician Abraham
Robinson came along and spotted a way round this argument. Why not simply accept
that infinitesimals aren鈥檛, in the technical sense, real? At least six times in
the history of mathematics, the meaning of the word number has been changed to
accommodate new variants. Even fractions and negative numbers were once
considered preposterous, but they proved indispensable and after a while no one
batted an eyelid. Mathematicians could likewise accept and use infinitesimals,
Robinson reasoned. There鈥檚 no need to pretend that they are real numbers.
So Robinson abandoned the Archimedean axiom and simply asked, what happens if
we accept that there are numbers too small to be expressed as decimals? He
worked out the arithmetic of such infinitesimals鈥攈ow to do maths with
them鈥攁nd found that unlike Newton鈥檚 paradoxical fluxions, they make
perfect sense.
You can add, subtract, multiply and divide infinitesimals, and even combine
them with real numbers to produce 鈥渘onstandard鈥� numbers. Adding an infinitesimal
to a real number gives you a finite nonstandard number鈥斺�漷hree and a
vanishingly small bit鈥�, for example. Dividing a real number such as 1 by an
infinitesimal produces infinity, as you might expect, but it is a specific
infinity that makes perfect logical sense.
What鈥檚 more, an infinitesimal avoids the contradiction of having to be
smaller than itself. It only has to be smaller than all non-zero real numbers,
and that鈥檚 not a problem as they aren鈥檛 real anyway.
What Robinson ended up with was a new and consistent number structure that
included all the usual real numbers and arithmetic operations, but supplemented
them with the infinitesimals and infinite numbers. This new branch of maths,
known as nonstandard analysis (NSA), cloaks each ordinary number with a cloud of
nonstandard numbers, all closer to it than any other real number and differing
from each other by an infinitesimal amount. It gives the real numbers an
infinitesimal fuzz.
Subtle knife
The upshot of Robinson鈥檚 discovery is that approaches like that of Archimedes
can be made rigorous. You really can find the volume of a sphere by adding
together infinitely many infinitesimal slices. It鈥檚 as though NSA gives you a
knife sharp enough to cut it up that small and handle the pieces. This is close
to our intuitive idea of why this method works. And if the test of a
mathematical model is how easily it helps us understand the world, then
infinitesimals pass with flying colours. In a way, it seems that the world
really is made of infinitesimal pieces.
The infinitesimals and their other nonstandard companions are now coming into
their own in many areas of mathematics and physics. Take the Boltzmann equation.
This describes how a cloud of tiny particles, such as the molecules in a gas,
changes in density as the particles move and collide. Physicists use computers
to solve the equation, to predict gas flows in interstellar clouds, for example.
The same equation can even be adapted to replace atoms with stars, and so
predict the motions of stars in a galaxy.
But the equation held a dark secret. No one had proved that it wouldn鈥檛 go
haywire somewhere. That is, you couldn鈥檛 be sure that for some new situation the
equation wouldn鈥檛 predict points of infinite density, or some other absurdity.
And if you can鈥檛 even rely on that much, you can鈥檛 trust numerical
approximations on a computer.
No one has yet found a 鈥渃lassical鈥� proof that the equation always has neat
solutions, because the huge numbers of particles that could be involved mean
there are too many possible situations to cover. But in 1984, Lars Arkeryd
decided to use infinitesimals instead. By making the molecules infinitesimally
small, he could use the mathematics of NSA to do the proof once and for all, and
he succeeded in proving that the equation is well behaved after all. Physicists
can now trust their simulations.
A case where NSA has actually given physicists new results is Brownian
motion鈥攖he apparently random movement of small bits of dust or smoke
particles caused by molecules of the surrounding fluid buffeting them. This kind
of unpredictable behaviour also describes the movement of prices in stock
markets, the movement of data in computer networks and many other situations, so
if you find a model for one you can apply it to all the related areas.
In the 1920s, Norbert Wiener of MIT discovered how to calculate some of the
characteristics of Brownian motion, by considering the average properties of the
fluid as if it were a continuous substance鈥攕moothing out the molecules
into a continuous goo. But Wiener鈥檚 approach was extremely technical. Worse, it
only revealed the general properties of Brownian motion. There is nothing in his
theory that corresponds to the trajectory of an actual bit of grit. It鈥檚 like
having a theory of the Solar System in which there are no planets and no
orbits.
In 1976, Robert Anderson discovered a much better and simpler approach to the
problem using NSA. He started by modelling the whole problem rather like a game
of 3D chess on a very large chessboard. The model chops space up into cubes,
some of which contain a piece that represents a molecule. At each tick of a
clock the pieces move randomly, either north, south, east, west, up or down.
Because this system is discrete, the calculations can be done
combinatorially鈥攖hat is, by counting things. This avoids all the
technicalities of Wiener鈥檚 approach, and it鈥檚 relatively easy to work out what鈥檚
going on.
Of course, real molecules move continuously rather than on a lattice. But you
can easily model this with infinitesimals: make your chessboard from infinitely
many infinitesimal squares. The successive moves are separated by a fixed, but
infinitesimal, interval of time. This way Anderson could work out real
trajectories for single dust particles, or single shares on the stock
market.
Much more recently, some of the oddest aspects of infinitesimals have been
appearing in pure maths. In 2000, Vladimir Kanove of Moscow University and
Michael Reeken of Wuppertal University in Germany published a nonstandard proof
of the Jordan curve theorem. This theorem states that every joined-up or
鈥渃losed鈥� curve divides the plane into two distinct regions, an inside and an
outside. It may sound obvious, but it鈥檚 not. Curves can be very
complicated鈥� for example, a spiral embellished with many smaller spirals
and so on forever. Proving the Jordan theorem for such a curve is anything but
straightforward.
It is straightforward, however, if the curve is a polygon. To tell if a point
is inside or outside, simply draw a straight line away from the point far enough
to get well away from the polygon. If the line crosses the polygon an odd number
of times then your point is inside. If it crosses an even number of
times鈥攕uch as zero鈥攖hen it is outside.
Kanove and Reeken have shown that you can approximate any closed curve by a
single polygon with infinitely many infinitesimal sides. This differs from the
original curve by an infinitesimal amount, but now you can use the same odd/even
argument to prove the theorem. The twist is that you might cross the polygon an
infinite number of times, but that鈥檚 alright. With NSA you can tell if an
infinite integer is odd or even: if you can make it by multiplying some other
nonstandard infinite integer by two, then it鈥檚 even. If you have to double
another infinity and then add one, it鈥檚 odd.
If all that is too abstruse for you, consider something more everyday:
computer graphics. A monitor screen is composed of finitely many tiny
rectangular pixels, but it can be modelled instead as a lattice with infinitely
many infinitesimal pixels. Jean-Pierre Reveill猫s of Lois Pasteur
University in Strasbourg has shown that this has many advantages, especially
when you want to rotate an image through some angle. In the ordinary case, when
you rotate a finite lattice of points through some angle, they don鈥檛 usually fit
very neatly into the original grid of pixels you have on your screen, so working
out the formula for how to do it is tricky. But if you instead work out how to
do the rotation for an infinity of pixels, then you can instantly adapt that
formula to rotate a finite number, instead of having to do the calculations from
scratch each time. Other manipulations can also be simplified this way, and
infinitesimals could one day be behind incredible special effects and computer
games.
And perhaps more than mere games. Juha Oikkonen of the University of Helsinki
reckons that NSA will be able to answer questions about what a computer can in
principle do. 鈥淚n theoretical computer science, one studies extremely
complicated finite situations,鈥� he says. The limits of computing power may be
related to the passage between the infinite and the finite, and Oikkonen鈥檚 guess
is that infinitesimals may be the way to explore this borderline. But for now it
is only a guess.
Admittedly, infinitesimals take a little getting used to. These slices of
nothingness might seem like nonsense at first, but in mathematics you should
never give up on a good idea just because it doesn鈥檛 make sense.