Chapter 2. Mathematical Cultures I
In Chapter 1 we looked at the origin of mathematics in the everyday lives of people.
The evidence for the conclusions presented there is indirect, coming from archae-
ology, anthropology, and other studies not directly mathematical. Wherever there
are written documents to refer to, we can know much more about what was done
and why. The present chapter is a broad survey of the development of mathemat-
ics that arose spontaneously, as far as is known, in a number of cultures around
the world. We are particularly interested in highlighting the motives for creating
mathematics.
1. The motives for creating mathematics
As we saw in Chapter 1, a certain amount of numerical and geometric knowledge is
embedded in the daily lives of people and even animals. Human beings at various
times have developed more intricate and sophisticated methods of dealing with
numbers and space, leading to arithmetic, geometry, and beyond. That kind of
knowledge must be taught systematically if it is to be passed on from generation
to generation and become a useful part of a civilization. Some group of people
must devote at least a part of their time to learning and perhaps improving the
knowledge that has been acquired. These people are professional mathematicians,
although their primary activity may be commercial, administrative, or religious,
and sometimes a combination of the three, as in ancient Egypt.
1.1. Pure versus applied mathematics. How does the mathematics profession
arise? Nowadays people choose to enter this profession for a variety of reasons. Un-
doubtedly, an important motive is that they find mathematical ideas interesting to
contemplate and work with; but if there were no way of making a living from having
some expertise in the subject, the number of its practitioners would be far smaller.
The question thus becomes: "Why are some people paid for solving mathematical
problems and creating new mathematical knowledge?" Industry and government
find uses for considerable numbers of mathematicians and statisticians. For those
of a purer, less applied bent, the universities and offices of scientific research subsi-
dized by governments provide the opportunity to do research on questions of pure
mathematical interest without requiring an immediate practical application. This
kind of research has been pursued for thousands of years, and it has always had
some difficulty justifying itself. Here, for example, is a passage from Book 7 of
Plato's Republic in which, discussing the education of the leaders of an ideal state,
Socrates and Glaucon decide on four subjects that they must learn, namely arith-
metic, geometry, astronomy, and music. These four subjects were later to form the
famous quadrivium (fourfold path) of education in medieval Europe. After listing
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