杏吧原创

NOTHING is more interesting than nothing, nothing is more puzzling than
nothing, and nothing is more important than nothing. For mathematicians, nothing
is one of their favourite topics, a veritable Pandora鈥檚 box of curiosities and
paradoxes. What lies at the heart of mathematics? You guessed it: nothing.

Word games like this are almost irresistible when you talk about nothing, but
in the case of maths this is cheating slightly. What lies at the heart of maths
is related to nothing, but isn鈥檛 quite the same thing. 鈥淣othing鈥� is鈥攚ell,
nothing. A void. Total absence of thingness. Zero, however, is definitely a
thing. It is a number. It is, in fact, the number you get when you count your
oranges and you haven鈥檛 got any. And zero has caused mathematicians more
heartache, and given them more joy, than any other number.

The history of the origin of zero is, appropriately, almost nonexistent.
Historical records are scanty, and those that do exist are open to innumerable
interpretations. Considering zero to be a number is different from having a
symbol for zero, and using a special symbol to indicate the absence of number is
different from having a symbol for zero. Indeed, the Babylonians had a special
symbol for the absence of a number in the Seleucid period, around 300 BC, but it
wasn鈥檛 a true zero symbol because they didn鈥檛 use it consistently. There is no
evidence to suggest that the Babylonians thought that 鈥渁bsence of number鈥� is a
kind of number, any more than we think that 鈥渁bsence of hair鈥� is a kind of
hair.

Zero, as a symbol, is part of the wonderful invention of 鈥減lace notation鈥�.
Early notations for numbers were weird and wonderful, a good example being Roman
numerals, in which 1998 comes out as MCMXCVIII鈥攐ne thousand (M) plus one
hundred less than a thousand (CM) plus ten less than a hundred (XC) plus five
(V) plus one plus one plus one (III). Try doing arithmetic with that lot. So the
symbols were used to record numbers, while calculations were done using the
abacus, piling up stones in rows in the sand or moving beads on wires.

Symbol status

At some point, somebody got the bright idea of representing the state of a
row of beads by a symbol鈥攏ot our current 1, 2, 3, 4, 5, 6, 7, 8, 9, but
something fairly similar. The symbol 9 would represent nine beads in any
row鈥攏ine thousands, nine hundreds, nine tens, nine units. The symbol鈥檚
shape didn鈥檛 tell you which, any more than the number of beads on a wire of the
abacus did. The distinction was found in the position of the symbol, which
corresponded to the position of the wire. In the notation 1998, for instance,
the first 9 means nine hundred and the second ninety. And so place notation was
born, probably in India, maybe with Arab help, not too long after AD 200.

The idea of place notation made it rather important to have a symbol for an
empty row of beads. Without it, you couldn鈥檛 tell the difference between 14,
104, 140, 1400, and so on. So, in the beginning the symbol for zero was
intimately associated with the concept of emptiness, rather than being a number
in its own right. However, by AD 800 things started to change when the Indian
mathematician Mahavira explained that multiplying a number by 0 produced 0 and
that subtracting 0 from a number left the number intact. By using 0 in
arithmetic on the same footing as the other numbers, he showed that 0 had
genuine numberhood.

Pandora鈥檚 box was now wide open, and what burst forth was鈥攏othing. And
what a glorious, untamed, infuriating nothing it was. The results obtained by
doing arithmetic with zero were often curious, so curious sometimes that they
had to be forbidden. Addition had the same effect as subtraction: the number
stayed the same. Linguistic purists may object that leaving something unchanged
hardly amounts to addition, but mathematicians generally prefer convenience to
linguistic purity. Multiplication by zero, as Mahavira said, always yielded
zero. It was with division that the serious trouble set in.

Dividing 0 by a non-zero number is easy: the result is 0. Why? Because 0
divided by 7, say, should be 鈥渨hatever number gives 0 when multiplied by 7鈥�, and
0 is the only thing that fits the bill. But what is 1 divided by 0? It must be
鈥渨hatever number gives 1 when multiplied by 0鈥�. Unfortunately, any number
multiplied by 0 gives 0 not 1, so there鈥檚 no such number. Division by zero is
therefore forbidden, which is why calculators put up an error message if you try
it.

Instead of forbidding fractions like 1 divided by 0, it is possible to
release yet another irritant from Pandora鈥檚 mathematical box鈥攂y defining 1
divided by 0 to be 鈥渋nfinity鈥�. Infinity is even weirder than zero; its use
should always be accompanied by a government warning: 鈥淚nfinity can seriously
damage your calculations.鈥� Whatever infinity may be, it isn鈥檛 a number in the
usual sense. So mostly it鈥檚 best to avoid things like 1 divided by 0.

Sorry: Pandora鈥檚 curse is not so easily evaded. What about 0 divided by 0?
Now the problem is not an absence of suitable candidates, but an embarrassment
of them. Again, 0 divided by 0 should mean 鈥渨hatever number gives 0 when
multiplied by 0鈥�. But since this is true whatever number you use to divide 0 by,
unless you鈥檙e very careful, you can fall into many logical traps鈥攖he
simplest such being the 鈥減roof鈥� that 1 = 2 because both equal 0 when they are
divided by 0. So 0 divided by 0 is also forbidden.

Alas, 0 divided by 0 was too seductive to stay forbidden for long. It is at
the heart of calculus, the independent invention of Gottfried Wilhelm von
Leibnitz and Isaac Newton. Calculus was an extraordinary intellectual
revolution, perhaps without historical parallel, because it gave us the idea
that nature is at root mathematical.

In what sense is calculus about 0 divided by 0? Well, the underlying feature
of calculus is the rate of change of some variable鈥攈ow rapidly it is
changing at a given instant. Here a formula or two seems unavoidable. Suppose
some quantity x varies with time t, and write x(t)
for its value at time t. This x might be how far your
bike has travelled, so x(12 noon) = the Pig and Whistle pub. At that
point your bike is probably not moving, unless an unfriendly local is nicking
it, so the rate of change of x at 12 noon is zero. However, by some
time t a bit after 2 pm, you are pedalling along the leafy byways at
position x(t). How fast is your position changing, at that
precise instant?

Newton鈥檚 answer was to let time increase by a tiny amount鈥攍et鈥檚 call it
d. As t changes to t + d, your bike moves from
x(t) to x(t + d)鈥攕ay from level
with a dozing sheep鈥檚 left nostril to level with its right nostril. The amount
by which your position changes is x(t + d)鈭抶
(t), the inter-nostril distance, and since it took you time d
to achieve that change, the rate of change is (x(t + d)
鈭� x(t))/d鈥攄istance travelled divided by
time taken to do so.

Mystery method

So far so good, but this expression represents the average rate of change
over the time interval from t to t + d, not the rate
of change at time t itself. However small d may be, even if
it鈥檚 0.00000000001, this approach still doesn鈥檛 give you the instantaneous rate
of change. Newton鈥檚 idea was to find the average rate of change over a time
interval of length d, let d become zero, and see what you
get.

In practice this leads to entirely sensible answers, but the procedure is
mysterious. Enter Bishop Berkeley, best known for his philosophical writings on
the problem of existence. Berkeley annoyed all the mathematicians by pointing
out鈥攃orrectly鈥攖hat Newton鈥檚 procedure amounts to dividing 0 by 0.
Over a time interval of zero, your bike moves a distance of zero, and you鈥檙e
dividing one by the other.

Berkeley had an ulterior motive: he was upset by criticisms that religious
faith was illogical, and he hit back by pointing out that calculus is illogical
too. He did so in a pamphlet entitled The Analyst, Or a Discourse Addressed
to an Infidel Mathematician Wherein it is examined whether the Object,
Principles, and Inferences of the Modern Analysis are more distinctly conceived,
or more evidently deduced, than Religious Mysteries and Points of Faith. It
contained the following: 鈥淔irst cast out the beam in thine own Eye; and then
shalt thou see clearly to cast out the mote out of thy brother鈥檚 Eye.鈥� Clearly
the good bishop was a bit peeved; equally clearly, he did his homework on the
maths.

Weasel words

Newton tried to justify his calculations by appealing to physical intuition,
and also by a rather weaselly explanation of how the method avoids dividing by
zero. First you write down your equation using the variable d. The
fraction involves dividing by d, but that鈥檚 all right because at this
stage you鈥檙e saying that d is not zero. You then simplify your fraction
until the d in the denominator disappears. Only then do you let d
equal zero to get your answer.

How d can sometimes be allowed to be zero and sometimes not, Newton
never really explained. Leibnitz made a more mystical appeal to the 鈥渟pirit of
finesse鈥� as opposed to the 鈥渟pirit of logic鈥� (which loosely translates as 鈥淚
don鈥檛 know what I鈥檓 doing, but hey, it works鈥�). Berkeley claimed the method
worked because of compensating errors, but missed the key point: why do the
errors compensate?

In the end, the whole problem was tidied up by Karl Weierstrass, about 120
years later, who defined the elusive concept of a 鈥渓imit鈥�. Rather than saying
that d sometimes can and sometimes cannot be zero, you鈥檙e actually
calculating the value that the fraction approaches as d gets closer and
closer to zero. And it works. So Newton and Leibnitz created a new way of
thinking about the world, while Berkeley鈥檚 criticism, though right, was
uncreative. The whole dispute, in fact, turned out to be about nothing.

In a sense, the whole of mathematics is about nothing, too, as John von
Neumann discovered. Von Neumann, a Hungarian who emigrated to the US in 1930,
came up with some of the most important new maths of the early 20th century,
including game theory, quantum logic and early steps towards the electronic
computer.

But his insights can also help us with the problem of what is a number. What,
for example, is two? It鈥檚 not the symbol 2鈥攁 symbol denotes a number, but
it鈥檚 not the same as the number. The logo 鈥淣ew 杏吧原创鈥� denotes the magazine
you are reading, but if you paid good money to your newsagent and received just
the words 鈥淣ew 杏吧原创鈥� written on a small card, you wouldn鈥檛 agree that the
symbol is the same as the reality. And it鈥檚 no good putting forward two cows or
two oranges鈥攖hose are examples of twoness, but not two itself. After all,
if you asked what 鈥済reen鈥� is and someone showed you a cabbage and an emerald,
you鈥檇 object that green is a colour, not a vegetable or a precious stone.

The answer turned out to be a clever modification of set theory by von
Neumann. Sets were the brainchild of Georg Cantor, a 19th-century mathematician
who was both brilliant and mentally unstable. A set is a collection of objects
which fit together because they have some element in common. Curly brackets like
these {} are used to denote the set formed by their contents. For example, the
set {mouse, cat, dog} is a collection containing the three named animals.

When mathematicians were trying to formulate the concept 鈥渘umber鈥� in a
rigorous way, they decided that you could standardise a number to be a set
containing certain fixed members. For instance, you might define the number
鈥渢hree鈥� to be {mouse, cat, dog} and 鈥渇our鈥� to be {mouse, cat, dog, goat}. But
what manner of beasts should the members be? Von Neumann realised that once you
introduced the number 0, you could 鈥渂ootstrap鈥� all the rest like this: 1 = {0},
2 = {0,1}, 3 = {0,1,2} and so on. Each number is determined by the previous
ones.

What about 0, then? That ought to be defined as a set with no members at all.
Cantor had already introduced such a set. It is called the empty set, and can be
defined as = {}. Now you can build all of mathematics from the empty
set鈥攆irst by defining the whole numbers following von Neumann鈥檚 scheme,
then by combining them in pairs to get fractions, and so on. With the help of a
few ideas from set theory, the whole of mathematics can be constructed from
nothing.

The saga does not end there. Whatever the mathematical concept, you can ask
what happens when key features are zero or empty: a space of zero dimensions is
a single point; a series with no terms adds up to 0 but multiplies to give 1;
zeroth roots make no sense at all. For many years there was a raging controversy
in graph theory, which describes the properties of networks of points joined by
lines, about whether the null-graph鈥攚ith no points and no lines鈥攊s
worth considering. The debate culminated in 1973 with a paper entitled 鈥淚s the
null-graph a pointless concept?鈥�. It all goes to show that nothing can cause
more trouble than nothing.

ZERO is a born troublemaker. Once mathematicians decided to treat it as a
number, all the standard formulas had to be extended to embrace zero鈥攚ith
results that were not always intuitive.

The most familiar example is powers. Take the fourth power of 5. This is 5⁴
or 5 脳 5 脳 5 脳 5. So clearly 5⁰ must be five to the power of zero, or no fives
multiplied together.

This is obviously not the way to think of it. Instead, what mathematicians do
is to decide on some property of powers that they want to remain true. For
instance, if you multiply powers together, the exponents add up:
5² 脳 5³ = (5 脳 5) 脳 (5 脳 5 脳 5) = 5⁵

If you want 5⁰ to be any use, it makes sense to retain this property, so
that 5⁰ 脳 5² must equal 5⁰⁺² = 5² = 25.
That is, 5⁰ 脳 25 equals 25.
Therefore 5⁰ must equal 1. For this reason, the standard convention is that the
zeroth power of any number is 1鈥攚ith one exception: 0⁰. The above
argument requires 0⁰ 脳 0 to equal 0, sure鈥攂ut now you can鈥檛 divide out
the 0s to conclude that 0⁰ = 1. In fact, just like 0 divided by 0, you have to
deem 0⁰ meaningless.

The same kind of approach determines the convention for zero factorial.
Factorials, symbolised by an exclamation mark, are normally defined like this:
5! = 5 脳 4 脳 3 脳 2 脳 1. Starting with the chosen number, reduce it one step at a
time until you hit 1, then multiply the resulting numbers together.

But this doesn鈥檛 help with 0!, since you have to stop before you start. The
usual interpretation of n! is 鈥渢he number of ways to arrange n
things in order鈥�, but this also doesn鈥檛 help, since it鈥檚 not at all clear how
many ways there are to arrange no things in order. The most plausible answer
would seem to be 鈥渘one, because there aren鈥檛 any things to arrange鈥�, but that
approach turns out to be misleading. Mathematicians prefer to preserve a general
property of factorials, the pattern

4! = 4 脳 3!

3! = 3 脳 2!

2! = 2 脳 1! and extend it to

1! = 1 脳 0! Since 1! = 1, this leads to the conclusion that 0! = 1. And this
is the convention that mathematicians employ.