A History of Mathematics From Mesopotamia to Modernity

46 A History ofMathematics

Octahedron

Cube

[Hexahedron]

Tetrahedron

Icosahedron Dodecahedron

Fig. 3The five regular (‘Platonic’) solids.

By ‘the incommensurability of side and diagonal’, Beckett means the fact, mentioned in our

discussion of theMeno, that the ratio of a square’s diagonal to its side,

√

2, is not a fraction. But

his pub classicists most probably came across their story not in its Greek original source, but in

its popularization in the twentieth-century history of mathematics, the ‘secret’ or ‘scandal’ of the

irrationals. This story, in some more respectable form, is still widely believed. The basic ‘fact’ is that

Pythagoras founded a sect of initiates whose secret knowledge was at least partly mathematical.

He is said to have taught that ‘all is number’, where by number is meant whole number (1, 2,

3,...); and at the same time he or his sect attached religious/magical importance to the regular

solids (see Fig. 3).

However, you cannot construct the regular solids, at any rate those which involve pentagons,

without bringing in irrational ratios—see Appendix B. Both this, and the problem of the side and

the diagonal, suggest a difficulty or, to put it more strongly, a ‘scandal’ for Pythagoras’s supposed

programme; because if ‘all is number’, then the ratio of the side to the diagonal should be the ratio

of two numbers.

There are stories in various late writers that this was kept secret by the Pythagoreans (supposedly

because it was a problem, though this is not explicitly stated), and that the secrecy was broken.

Iamblichus, a commentator of the late third centuryce(seven centuries after the events), refers

to Hippasos of Metapontum as a member of the Pythagoreans who was expelled and drowned at

sea (or some similar fate) for revealing a secret—in one version, the construction of one of the

solids, and in another, the nature of the rational and irrational. How far you accept this story, as

Beckett’s characters did, depends on your estimate of Iamblichus as a source, and he does not go

out of his way to inspire confidence. The next step for twentieth-century historians was to deduce

that the secret or scandalous nature of the irrationals for the Pythagoreans extended to the Greek

mathematical community in general; and that this accounted for their avoidance of measurement.

The definitive version of this story was due to Hasse and Scholz in the 1920s. Speaking of a ‘crisis

of foundations’ for Greek mathematics, they used it to explain Euclid’s use of proportions:

Given that the Greeks were born geometers, as they are usually held to be, it must be concluded with certainty that

after such a foundational crisis they needed to construct a purely geometric mathematics. In such a mathematics we