The Age of Computation
ideas in this article come from Douglas Campbell, The
Whole Craft of Number
(Boston: Prindle, Weber, &
Imagine living before -600 with a nomadic tribe and spending most
of your time hunting or herding. Occasionally you
compare your flock of sheep with a pile of pebbles
to be sure that no sheep have been lost. Maybe you
have names for the numbers we call one, two, and three,
but more than three you call many. You probably
have a map in your head of where important geographical
landmarks are located relative to each other. Sometimes
the shaman of your tribe draws pictures in the dirt
representing the sun and moon and elk, to bring about
good weather or happy hunting. You and your
neighbors represent reality through some simple symbols:
stones, maps, pictures.
Every cloudless night there is this fantastic light
show. In the summer you go out and lie in the grass
and stare for hours up at the big screen in the sky.
Most of the show is pretty much the same from night
to night; the points of light move in the same way
through the evening. You never tire of finding shapes
among them. Perhaps you imagine a hunter
chasing animals across the heavens. There
are a few lights, though, that move differently from
the others. Some nights they'll be in one part of the
sky and some in another. You call these lights the
wanderers, or planets.
You also find that you can create beauty in paintings
and by decorating pottery. You discover how different
shapes make objects more beautiful. You especially
like shapes that are symmetric. And perhaps you begin
to phrase in your head an important problem:
The Symmetry Problem:
What is symmetry?
This is a problem mathematicians will work on
The Size Problem
Or imagine that you live in an agricultural community.
You enjoy looking at the sky, too. You have names
for larger numbers, also, since they're needed for trading.
You know some recipes for making complicated foods
like bread. You compare quantities of objects, and
you have a vague notion of relative sizes of fields
and volumes of harvests. Instead of one stone for
each sheep, you save space by replacing five regular
stones with one special red stone, and five red stones
with a white stone. And the shaman of your tribe actually
draws pictures representing the stones, which were
representing the sheep. That is, your tribe has symbols
to represent symbols.
You are interested in the sizes of plots of agricultural
land. And you want to be able to compare those sizes.
You phrase a second major problem of mathematics:
How can we find the size of any shape?
Little do you imagine that this problem will be a major
question of the branch of mathematics called calculus.
Or imagine living later, as part of a larger political
system. Such large collections of people, united by
conquest, existed in China, India, Egypt, Mesopotamia,
and Central America. They had alphabets, ships, and
metals, creating an economy based on trading, and not
just trading agricultural products.
You are part of the huge bureaucracy that runs
the system. At the top might be an Emperor. Your
boss is the local viceroy. Your job is to provide
your boss with information. You use a system that
systematically gives names to large numbers, so that
the population can be counted. To levy taxes you compute
the areas of pieces of land and do arithmetic with
the resulting numbers. You work with geometric forms
to help design large buildings for the Emperor. To
make recommendations about when planting should occur
and when rivers will flood, you must know about weather
patterns and the calendar. And sometimes you just
don't explain how you figure out some things. People
seem to give you more respect if they think that your
god told you when the winter solstice would be coming.
Moreover, if you announce that your god is unhappy
and needs appeasing, people can be persuaded to give
If you live in Mesopotamia, you have access to
lists of information compiled by your predecessors.
If the viceroy asks you for the cost of outfitting
a crew of slaves, workers, and supervisors to build
a building, you can look up in a table the numbers
of each kind of person and read off the cost without
doing a lot of calculation. The computations have
been done in advance.
You even have some time to mess around with less practical
matters. For example, you once made a list of numbers
describing the lengths of sides of various right triangles.
Others have made tables of very good fractional approximations
to square roots. Still others have listed approximate
solutions to many equations. Some of your colleagues
who are really good at measuring the number of degrees
in an angle make long lists of the angles that describe
where those wanderers are in the sky.
If you live in Mesopotamia, then your number system
is based on the number 60. If you are in China, you use a system like ours
with base 10, but again not with consistent positional
notation. Some early Mayan city states in North and
Central America used a positional number system with
base 20 and also had a symbol for zero. The dates
of these developments are not known.
What mathematics don't
you and your colleagues do? You
don't abstract. That is, you might be able to solve,
in effect, lots of specific quadratic equations, but
you don't represent the numbers abstractly (by letters)
to come up with general formulas. You also don't distinguish
between exact and approximate answers; for example,
calculating the circumference of a circle by multiplying
the diameter by 3 is close enough for government work.
That is, unless you live in Egypt, where you have a
pretty good system of fractions, now called Egyptian fractions
To find the area of the circle
you'd multiply the square of the radius by
which today we call 49/16. (We recognize that
49/16 is an approximation to the number we call ¼.)
As a bureaucrat, scientist, mathematician, and priest
rolled into one, you do computations primarily for
the state. But after work, you go out and stare at
the sky. Maybe there's a pattern in the movement of
Boyer, Carl B., A History of Mathematics (New York:
John Wiley & Sons, 1968).
Campbell, Douglas, The Whole Craft of Number (Boston:
Prindle, Weber & Schmidt, 1976).
Eves, Howard, An Introduction to the History of Mathematics
(Chicago: Saunders College Publishing, 1990).