PERMUTATIONS 29

CAUTION: You should read the section on permutations

several times if necessary, because it is the basic idea that

makes Galois theory come to life. We have used

{a,b,c}

as our

basic example. You could play around with

A

where A has 1,

2, 4, or 100 elements. If A has 100 elements, do not try to

write out all the elements of

A

. There are 100! of them, and

it would take you almost 3 × 10

150

years if you wrote out one

element every second.

Digression: Mathematics and Society

Mathematics is like a game. It has rules, and to enjoy playing

or watching it, you have to know and understand the rules.

Mathematicians make up the rules as they go along. There is

sometimes an extrinsic beneﬁt, because mathematics is used for

many practical things, starting with counting and telling time,

and including theoretical physics and constructing computers. Even

number theory has extrinsic beneﬁts nowadays, coming mostly

from the theory of codes and ciphers, but also in acoustics, radar,

and other areas. But even when no applications are immediately

apparent, playing the game can be satisfying.

Many people believe that all of mathematics has already been

discovered and codiﬁed. Mathematicians (they think) do nothing

except rearrange the material in different ways for different types

of students. This seems to be the result of the cut-and-dried method

of teaching mathematics in many high schools and universities.

The facts are laid out in the cleanest logical order. Little attempt

is made to show how someone once had to invent it all, at ﬁrst in

a confused way, and that only later was it possible to give it this

neat form. Many textbooks make no effort to tell about directions

that are still to be explored, conjectures that are unproven, nor, of

course, of ideas that are yet to be formulated.

Project yourself back in time to 1000

BC. Very little mathematics

was known then. It all lay in the future to be discovered, debated,

arranged, and improved. The situation today is nearly the same!

30 CHAPTER 3

Very little has been discovered until now compared with the

amount yet to come.

If you read books such as this one or the articles in the

newspapers about mathematics, you know that new discoveries,

sometimes very important ones, are made from time to time. But

newspapers cannot report on the entire web of ideas and proofs

that is continually being extended by mathematicians around the

world. The same is true, of course, in the other sciences.

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