From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 267
distance of ships at sea requires the use of this theorem. (352.14–18, all four
passages in DK 11 A 20)
Other passages 24 credit Thales with a method for determining the height of a
pyramid by measuring its shadow and call him ‘the first to describe the right
triangle of a circle’, whatever that may mean. However, the crucial passages are
the four I have quoted. Dicks ([8.88], 302–3) seizes on the last to argue that
Eudemus’ attributions to Thales are reconstructions which presuppose that
Thales demonstrated in a basically Euclidean way geometrical theorems implicit
in his more practical accomplishments. Even if one accepts the plausibility of
this approach to ancient doxographical reports, two features of these particular
ones may cause one to hesitate: the detailed point in (b) about Thales’ archaic
vocabulary, and the fact that in (a) Thales is said to have proved something
which is (illegitimately) made a matter of definition in Euclid’s Elements (I, def.
17).
It is also striking that all four of the propositions ascribed to Thales can be
‘proved’ either by superimposing one figure on another (d) or by ‘folding’ a
configuration at a point of symmetry. It seems possible that Thales’ proofs were
what we might call convincing pictures involving no explicit deductive structure.
But once one ascribes even this much of a conception of justification to Thales,
one is faced with what would seem to be serious questions. How did it come
about that Thales would formulate, say, the claim that a diameter bisects a circle?
If he was just interested in the truth ‘for its own sake’, then we already have the
idea of pure geometrical knowledge. But if, as seems more plausible, he was
interested in the claim as a means to justifying some other less obvious one, then
we seem to have the concept of mathematical deduction, from which the
evolution of the concept of mathematical proof is not hard to envisage. We need
not, of course, suppose that Thales was a rigorous reasoner by Euclidean
standards; merely saying that he explicitly asserted and tried to justify
mathematical propositions of a rather elementary kind is enough to give us a
primitive form of mathematics.
Of course, we would like to know something about the historical background
of Thales’ interests. Proclus and other ancient sources give credit to the Egyptians,
but modern scholars tend to be sceptical about these claims. 25 Van der Waerden
and others have invoked the Babylonians to fill the gap. I quote from his
discussion of Thales ([8.13], 89), which shows that he also credits Thales with a
high standard of mathematical argumentation.
We have to abandon the traditional belief that the oldest Greek
mathematicians discovered geometry entirely by themselves and that they
owed hardly anything to older cultures, a belief which was tenable only as
long as nothing was known about Babylonian mathematics. This in no way
diminishes the stature of Thales; on the contrary, his genius receives only
now the honor that is due it, the honor of having developed a logical