A History of Mathematics- From Mesopotamia to Modernity

Greeks and‘Origins’ 35

whole number, or even a ‘rational’ number (a fractionp/q, wherep,qare whole numbers). You can

approximate it as closely as you like by fractions, but the result will never be exact.

As Michel Serres points out, in a detailed discussion of the dialogue:

Nobody asks the asker: how long? He questions the ignorant slave about a content about which nobody, however,

bothers him. He found the side all right, but he did not measure it. Socrates is cheating: he knows that he will not find

the exact length. (Serres 1995, p. 105)

In fact, one reason why this particular problem has been chosen for the dialogue is perhaps—but

here, as usual, we have to start attributing motives to the Greeks—because it shows the limitations

of numbers, and the superior power of geometrical methods. The boy will never arrive at the

right answer by guessing different numbers; but Socrates can draw a picture which solves it. The

philosophers’ mathematics not only uses a more abstract idea of ‘number’, but when number

becomes a problem, it can dispense with it.^1

Greeks said that ratios such as diagonal-to-side were ‘alogoi’—without a reason, irrational. More

simply, they said that the side and the diagonal were ‘incommensurable’, that is, there is no shorter

linelhaving the property that both side and diagonal are exact multiples ofl(are ‘measured’ byl).

We shall return to these terms, and the problems they pose, later.

Exercise 1.Is it obvious that the figure which Socrates constructs in Fig. 1 is a square?Why?

Exercise 2.Why does ‘diagonal is incommensurable with side’ mean the same to us as ‘

√

2 is not a

fraction’?

Exercise 3.Why is

√

2 not a fraction anyway?

2. Literature

It is striking that near the end of the twentieth century there should appear two books arguing that much of the history

of Greek mathematics written during that century is wrong. Reviel Netz argues that it is wrong because historians

have not understood the crucial roles that language and diagrams played in shaping the deductive structure that is

Greek mathematics’ most striking characteristic. David Fowler argues that it is wrong because a key component of

the mathematics that developed in and around Plato’s academy was lost in Hellenistic times and was not rediscovered

until the Renaissance. (Berggren 2003)

The literature on ancient Greek mathematics, as the above quotation reminds us, is large and in
constant flux. We shall consider the specific problem of the ancient Greek texts themselves shortly.
For the moment, let us concentrate on what material is available on the period, as primary and sec-
ondary sources. The standard reference text is certainly Heath’s (1981), a reprint of a 1921 classic.
Because it is so old, and so much a standard work, it is the basis for most later authors’ arguments,
disagreements, and conjectural reconstructions. The mainprimarysources for the period we are
considering—up to and including Euclid, around 300bce—are the works of Plato, Aristotle, and
Euclid himself. Fauvel and Gray give plentiful extracts from all of them, and all can be found easily
on the Internet. It is very strongly recommended that you read some of these texts, which vary from

1. There is evidence that Plato did know that the side of the eight-foot square was not a rational number, from theTheaetetus—see
   extract in Fauvel and Gray 2.E.3, pp. 73–4. But the question is not raised in theMeno. Why not?