原文在這兒： https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf

THE IMPORTANCE OF MATHEMATICS
W. T. Gowers
It is with some disbelief that I stand here and prepare to address this gathering on the
subject of the importance of mathematics. For a start, it is an extraordinary honour to
be invited to give the keynote address at a millennium meeting in Paris. Secondly, giving a lecture on the significance of mathematics demands wisdom, judgment and maturity, and there are many mathematicians far better endowed than I am with these qualities, including several in this audience. I hope therefore that you will understand that my thoughts are not fully formed: if I had been asked to speak on this subject five years ago, I would have given a completely different lecture, and I am confident that in five years’time it would again have changed.
My title (which I did not actually choose myself, though I willingly agreed to it) also
places on me a great burden of responsibility. After all, I am speaking to an audience which contains not just mathematicians but journalists and other influential non-mathematicians.
If I fail to convince you that mathematics is important and worthwhile, I will be letting
down the mathematical community, and also letting down Mr Clay, whose generosity has made this event possible and is benefiting mathematics in many other ways as well.
Unfortunately, if one surveys in a superficial way the vast activity of mathematicians
around the world, it is easy to come away with the impression that mathematics is not
actually all that important. The percentage of the world’s population, or even of the
world’s university-educated population, who could accurately state a single mathematical theorem proved in the last fifty years, is small, and smaller still if Fermat’s last theorem is excluded. If you ask a mathematician to explain what he or she works on, you will usually be met with a sheepish grin and told that it is not possible to do so in a short time. If you ask whether this mysteriously complicated work has practical applications (and we all get asked this from time to time), then there are various typical responses, none of them immediately impressive.
One is the line taken by the famous Cambridge mathematician G. H. Hardy, who
was perfectly content, indeed almost proud, that his chosen field, Number Theory, had
no applications, either then or in the foreseeable future. For him, the main criterion of
mathematical worth was beauty. At the other end of the spectrum there are mathematicians who to work in areas such as theoretical computer science, financial mathematics or statistics, areas of acknowledged practical importance. Mathematicians in these areas can point to ideas that have had a big impact, such as the Black-Scholes equation for derivative pricing, which has transformed the operation of the financial markets, and the public-key cryptosystem invented by Rivest, Shamir and Adelman, which is now the basis for security on the internet, and which, as has been pointed out many times, is an application of number theory that Hardy certainly did not expect.
Also at the applied end of the spectrum there are many mathematicians whose work
has intimate connections with theoretical physics. Actually, it is not obvious that unifying General Relativity and Quantum Mechanics would have direct practical applications, since today’s physics already provides us with predictions that are accurate to within the limits we can measure. But one never knows, and in any case such a breakthrough would be of absolutely fundamental interest to science, or indeed anybody with the slightest intellectual curiosity. If mathematicians can make a contribution to this area, then they will at least be able to point to a huge external application of mathematics.
Most mathematicians, including me, lie somewhere in the middle of the spectrum,
when it comes to our attitude to applications. We would be delighted if we proved a
theorem that was found to be useful outside mathematics - but we do not actively seek to do so. Given the choice between an interesting but purely mathematical problem and an uninteresting problem of potential benefit to engineers, computer scientists or physicists, we will opt for the former, though we would certainly feel awkward if nobody worked on practical problems.
Actually, this attitude is held even by many of those who work in more practical-
seeming areas. If you press such a person, asking for a specific example of an application in business, industry or science of their own work as opposed to an application of a result in their general area, you will often, though not invariably, witness an uncomfortable reaction.
It turns out that a great deal of the research in even the so-called practical areas is in fact not practical at all. I am not trying to draw attention to some sort of scandal by saying this - as I hope to demonstrate, this phenomenon is a natural and desirable consequence of what it means to view the world mathematically.
The reason for it is that mathematics is a two-stage process. Rather than studying the world directly, mathematicians create so-called models of the world, and study them.

This applies even to the simplest mathematics. After the age of four or five we do not
study addition by actually combining groups of objects and counting them. Instead we
use an abstract mathematical construction, or model, known as the positive integers (that
is, the numbers 1,2,3,4,5 and so on). Similarly, we do not do basic geometry by cutting
shapes out of paper, partly because it is not necessary and partly because in any case the
resulting shapes would not be exact squares, triangles or whatever they were supposed to
be. Once again, we study a model, a sort of idealized world that contains things that we do
not come across in everyday life, such as infinitely thin lines that stretch away to infinity,
or absolutely perfect circles, and does not contain untidy, worldly things like hamburgers,
chairs or human beings.
If one works in a practical area of mathematics, then there will be two conflicting
criteria for what makes a good model. On the one hand, the model should be accurate
enough to be useful, and on the other, it should be simple and elegant enough to generate
realistic and interesting mathematical problems. It is tempting, as a mathematician, to
attach far more importance to the second criterion - mathematical interest and elegance -
than to the first - accuracy - even if this means not immediately contributing to the gross
national product of one’s country.
A good example of this attitude comes from computer science. Consider the network
shown in Figure 1, and imagine that we have been asked to colour the nodes of the network
with two colours, red and blue, in such a way that no two nodes of the same colour are
ever linked. Such a colouring is called a proper colouring of the network.
If we start with a single node, such as the one marked A, then it doesn’t matter
whether we colour it red or blue, since the roles of the two colours are interchangeable.
However, once we have decided that A should be red, say, then the two nodes marked B
have to be blue, since they are linked to A and therefore must not have the same colour as
A. Having established this, we then see that the nodes marked C must all be coloured red
(since they are linked to blue nodes), and then that all nodes marked D must be coloured
blue. But now we have hit a problem, which is that two of the nodes marked D are linked
to one another. Since all our choices of colour were forced from the moment we coloured
the node A, it follows that it is impossible to give a proper colouring of the nodes with
only two colours.
Now this argument does more than merely establish that one particular network can-
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not be properly coloured. We have used a general procedure, or algorithm, as it is called in
mathematics and computer science, for testing whether any given network can be properly
coloured with two colours. Briefly, this procedure can be described as follows: colour one
node arbitrarily and then continue by colouring nodes whenever the choice of colour is
forced. If you are eventually forced to give two linked nodes the same colour then the
network cannot be properly coloured, and if you are not then it can.
This procedure is sufficiently well determined that a computer can easily be pro-
grammed to carry it out. Obviously, the larger the network, the longer the procedure
takes, and hence the longer the computer will take to run the program. A careful analysis
shows that if the network has n nodes, then the number of steps needed by the computer is
proportional to n2. (It may seem to be only linear, but this is because for a small network
represented visually one can see at a glance where the neighbours of any given node are.
For a large network encoded as a string of bits this is no longer the case.) To give some
idea of what this means, if the network has 100 nodes, then the number of computer steps
will be around 10,000, and if it has 1,000 nodes then the number of steps rises to around
1,000,000.
Now let us modify our problem slightly. Figure 2 shows a new network. It can be
shown quite easily that this network cannot be properly coloured with two colours (for
example, consider the triangles towards the left of the network), but what happens if we
allow ourselves three colours? Is there a proper colouring?
This question turns out to be much harder, in general. The reason is that if one tries
to colour the network by colouring nodes in turn, then many of the choices one makes
are not forced in the way that they were with two colours. For example, if one wishes to
colour a node which is linked to two other nodes both of which have been coloured red,
one can do so with either green or blue. Then, if one runs into difficulty later, it may be
that this difficulty was an indirect consequence of this bad decision made much earlier. To
make things worse, there may have been several other decisions, of which some were bad
and some good, with no easy way to tell which was which. As a result, it is very hard to
establish conclusively that a proper colouring with three colours does not exist, and if it
does exist it can be very hard to find.
One method we could try is simply to examine all possible choices of colours and see
if one of them works. Of course, this would be very tedious, but isn’t tedious repetitive
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calculation what computers are good at? The answer is yes, but only if the number of
repetitions is not too large. For this problem, the amount of time needed by the computer
would be prohibitively large. If a network has n nodes, then the number of ways of assigning
each node one of three colours is 3n, and for each assignment of the colours it takes the
computer about n2 steps (at worst) to check whether there are two linked nodes of the
same colour. Hence, the number of steps needed by the computer is something like 3n.n2.
Again, one can understand what this means by considering a specific value of n such as
100. For a network with 100 nodes, the number of steps needed is 3100.1002, or
5153775207320113310364611297656212727021075220010000.
It would take all the world’s computers combined far longer than the universe has existed
to perform this number of steps.
It follows that a second procedure I have just outlined, namely check all possible
colourings and see if one of them is a proper colouring, is impractical in the extreme. As
it happens, there is no known practical way of determining whether a general network can
be properly coloured with three colours. However, you may be interested to know that the
network in Figure 2 can be: see Figure 3.
Obviously, if one is programming a computer, it is worth knowing whether one’s
program is likely to run in a reasonably short time. Hence, it is important in theoretical
computer science to establish what one means by a practical algorithm, or procedure. The
most widely used convention is that an algorithm is regarded as efficient if the number
of steps is no worse than a polynomial function of the size of the input. For example, if
somebody were to find a way of determining whether a network with n nodes could be
properly coloured with three colours using no more than 100n8 + 73n6 + 12n3 + n + 1
computational steps, then this would be regarded as the first practical solution of the
problem and a breakthrough of the first importance.
However, a computer programmer could make only limited use of such a breakthrough.
For example, when n = 100,
100n8 + 73n6 + 12n3 + n + 1 = 1000073000012000101
which is still far too many steps for a computer. So the practical solution would only be
‘practical in theory’ as opposed to ‘practical in practice’.
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The point I wish to make is that theoretical computer scientists have a notion of
practicality of algorithms which is different from genuine practicality, though related to
it. The reason they use this notion is that for mathematical reasons it is natural, elegant
and convenient. From a mathematical point of view, this definition of practicality leads to
questions which are often more interesting than those that arise from the genuine needs of
computer programmers. Thus, even in a very practical subject, practical problems are not
always of the highest priority. (I should add that theoretical computer scientists are well
aware of what I am saying, and are not indifferent to questions of genuine practicality.)
To summarize what I have said so far, most mathematicians, including those who work
in useful-sounding branches of mathematics, do not work on problems with direct practical
applications. It would be dishonest of me to argue for the importance of mathematics by
trying to pretend that this was not so. Instead, my task will be to explain why, despite
this fact, mathematics is a worthwhile endeavour, and why it should be supported. I will
give two arguments, the first based on the practical utility of mathematics (despite what
I have just said) and the second on its cultural value.
It may look as though I have been trying to convince you that mathematics is a useless
subject, but in fact all I have claimed is that a typical mathematician does not actively
try to be useful. These are two very different statements. They are different because there
is an important distinction between the collective result of an activity and the individual
motives of the participants. Let me give an example of this from outside mathematics.
Some capitalist economies are based on the premise that individual greed and selfishness,
to use somewhat emotive terms, can act for the collective good of society. The greed
causes people to strive to become wealthy, and this benefits the entire economy in many
ways, such as increasing the tax revenue for the government, which can then be spent on
hospitals, schools, public transport and so on, or causing companies to be set up, which
provide livelihoods for many people. The individuals need have absolutely no interest in
whether other members of society have satisfactory lives, provided that sufficient social
order is maintained, but in an indirect way their activity does benefit others.
Of course, not everybody agrees that economies such as I have briefly described are a
good thing, and the last thing I wish to do is let politics intrude on a mathematical lecture.
However, it is surely not controversial to state that individual selfishness can lead to public
good, and I wish to argue that something similar happens in mathematics. Although
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individual mathematicians are motivated primarily by a subtle mixture of ambition and
intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a
whole does benefit society. I would now like to examine in more detail why this is.
One straightforward answer is this: mathematics is cheap, and occasionally produces
breakthroughs of enormous economic benefit, either directly, as in the case of public-key
cryptography, or indirectly, as a result of providing the necessary theoretical underpinning
for science. If you were to work out what mathematical research has cost the world in
the last 100 years, and then work out what the world has gained, in crude economic
terms, then you would discover that the world has received an extraordinary return on a
very small investment. And I haven’t even mentioned the fact that those who engage in
mathematical research also teach very bright students, many of whom do not themselves
become mathematicians, but rather use their mathematical training in ways that directly
contribute to the world economy.
Taken as a whole, then, mathematics is undeniably important. However, a cost-cutting
finance minister will notice a gap in the above argument: might it not be possible to achieve
the same benefits more cheaply? If the benefits of mathematics come from teaching and
a few breakthroughs, while most mathematicians get on with their interesting but useless
research, then why not cut the research funding to the useless areas and just support the
teaching and the more practically oriented mathematics? One of my main objectives today
is to expose the fallacy, or rather fallacies, that would lie behind such a proposal.
The first one is the idea that it is possible to identify the areas of mathematics that
will turn out to be useful. In fact, it is notoriously hard to predict this, and the history of
mathematics is littered with examples of areas of research that were initially pursued for
their own sake and later turned out to have a completely unexpected importance. I could
mention the RSA algorithm yet again. A more fundamental example is the non-Euclidean
geometry of Gauss, Bolyai and Lobachevsky, which is internally consistent despite such
apparently paradoxical phenomena as the existence of triangles with angles not adding to
180o. This paved the way for Riemannian geometry, which seemed to be an example of
pure mathematics par excellence until it turned out to be exactly what Einstein needed
for his general theory of relativity.
A recent and celebrated example is provided by the theory of knots. Figure 3 shows
seven examples of what mathematicians call knots. These differ from ordinary knots in
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that the ends of the knotted string are fused together - perhaps ‘knotted loops’ would be
a more accurate term. A particularly simple knot is shown at the bottom. This is known
as the unknot, because, as one can easily see, it can be untwisted into a simple loop. The
other knots are more genuinely knotted, but this is surprisingly hard to prove, and to this
day there is no known practical method (that is, algorithm again) of deciding whether or
not a more complicated diagram of the kind appearing in Figure 3 represents a knot that
can be untied into a simple loop. Another central problem in knot theory is to decide
when two diagrams in fact represent the same knot, in the sense that one can be twisted
into the other. For example, though it is not obvious from the diagram, a good mental
gymnast can eventually see that the fourth and fifth diagrams (reading from the top, and
from left to right) represent the same knot.
Once again, these looked like amusing puzzles until, after work of Vaughan Jones
and Edward Witten, it was realized that knot theory had fundamental connections with
theoretical physics.
So - mathematicians can tell their governments - if you cut funding to pure mathe-
matical research, you run the risk of losing out on unexpected benefits, which historically
have been by far the most important.
However, the miserly finance minister need not be convinced quite yet. It may be very
hard to identify positively the areas of mathematics likely to lead to practical benefits, but
that does not rule out the possibility of identifying negatively the areas that will quite
clearly be useless, or at least useless for the next two hundred years. In fact, the finance
minister does not even need to be certain that they will be useless. If a large area of
mathematics has only a one in ten thousand chance of producing economic benefit in the
next fifty years, then perhaps that at least could be cut.
You will not be surprised to hear me say that this policy would still be completely
misguided. A major reason, one that has been commented on many times and is implied
by the subtitle of this conference, “A Celebration of the Universality of Mathematical
Thought”, is that mathematics is very interconnected, far more so than it appears on the
surface. The picture in the back of the finance minister’s mind might be something like
Figure 4. According to this picture, mathematics is divided into several subdisciplines, of
varying degrees of practicality, and it is a simple matter to cut funding to the less practical
ones.
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A more realistic picture, though still outrageously simplified, is given in Figure 5.
(Just for the purposes of comparison, Figure 6 shows Figures 4 and 5 superimposed.) The
nodes of Figure 5 represent small areas of mathematical activity and the lines joining them
represent interrelationships between those areas. The small areas of activity form clusters
where there are more of these interrelationships, and these clusters can perhaps be thought
of as subdisciplines. However, the boundaries of these clusters are not precise, and many
of the interrelationships are between clusters rather than within them.
In particular, if mathematicians work on difficult practical problems, they do not do so
in isolation from the rest of mathematics. Rather, they bring to the problems several tools
- mathematical tricks, rules of thumb, theorems known to be useful (in the mathematical
sense), and so on. They do not know in advance which of these tools they will use, but they
hope that after they have thought hard about a problem they will realize what is needed to
solve it. If they are lucky, they can simply apply their existing expertise straightforwardly.
More often, they will have to adapt it to some extent. Perhaps it will be helpful if I show
another two pictures, this time illustrating what it is like to solve a mathematical problem.
Figure 7 is a naive view: you start at the boundary of what is known, with a definite
goal in mind. You then have a succession of brilliantly clever ideas after which the solution
pops out. A more realistic view, which I have tried to represent pictorially in Figure 8,
takes into account numerous important features of mathematical research such as false
starts, promising ideas that lead nowhere or insights that unexpectedly solve a different
problem.
Thus, a good way to think about mathematics as a whole is that it is a huge body of
knowledge, a bit like an encyclopaedia but with an enormous number of cross-references.
This knowledge is stored in books, papers, computers and the brains of thousands of
mathematicians round the world. It is not as convenient to look up a piece of mathematics
as it is to look up a word in an encyclopaedia, especially as it is not always easy to
specify exactly what it is that one wants to look up. Nevertheless, this “encyclopaedia” of
mathematics is an incredible resource. And just as, if one were to try to get rid of all the
entries in an encyclopaedia, or, to give a different comparison, all the books in a library,
that nobody ever looked up, the result would be a greatly impoverished encyclopaedia or
library, so, any attempt to purge mathematics of its less useful parts would almost certainly
be very damaging to the more useful parts as well.
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So far I have simply stated that mathematics is full of surprising connections. Any
mathematician will happily confirm that statement and be able to give examples from
his or her experience. Indeed, discovering surprising connections is one of the great joys
of the subject. I would like to illustrate the interconnectedness of mathematics with an
example in which I played a small part. To do this I must briefly describe a few unsolved
mathematical problems.
The first one is simple enough that I can explain it precisely. Consider the following
three sequences of numbers:
5 11 17 23 29
7 19 31 43
11 41 71 101 131
They have two important features in common. First, they go up in regular steps: thus, the
first one goes up by 6 each time, the second by 12 and the third by 30. Such a sequence
is called an arithmetic progression. Secondly, and more interestingly, they all consist only
of prime numbers.
If you try to extend one of these sequences in the natural way, then it will no longer
consist solely of primes. For example, 29 + 6 = 35, which is 5 × 7. Similarly, 55 = 5 × 11
and 161 = 7 × 23. In fact, it is relatively straightforward to show that any arithmetic
progression, if continued far enough, contains numbers that are not prime. However, this
observation still leaves open the following two questions:
Problem 1. Are there infinitely many arithmetic progressions of length four consisting
of prime numbers?
Problem 2. Can arithmetic progressions of primes have any (finite) length or is there
some upper limit to how long they can be?
I should say in passing that if you replace ‘four’ by ‘three’ in question (1), then the answer is
known to be yes, but the proof is by no means easy. Secondly, the longest known arithmetic
progression consisting of prime numbers has length in the early twenties and was found by
a computer. Of course, no such example leaves us any the wiser about question (2).
The next problem I wish to describe is more geometrical in character. Figure 9a shows
a triangle divided up into eight tall thin triangles. This is done in the most obvious way:
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