Greeks and‘Origins’ 45
Pythagoras at all.^8 That he did exist is a reasonable assumption, since he is referred to by Herodotus
and Heraclitus not long after his time; but not much more can be said. With reasonable evidence
that a revolution of some sort took place, we have no serious information on the state of affairs
before or on what happened; only some idea of the situation some time after. Netz makes an
interesting case for dating this ‘first revolution’ to the (rather late) time of Hippocrates, about
440 bce:
According to our evidence, mathematics appears suddenly, in full force. This is also what one would expect ona priori
grounds. I therefore think mathematics, as a recognizable scientific activity, started somewhere after the middle of the
fifth century B.C. (Netz 1999, p. 275)
His arguments are persuasive, but in the nature of things can hardly be conclusive.
However, as we have seen, there are more unusual things to be accounted for in Euclid, and
in much of what followed, than the use of argument and diagram; and the introduction of such
special features, which we could provisionally think of as the second revolution, is the one which
has particularly attracted the imaginative historians. The evidence is still a mixture of gossip
and inventive reconstruction, but there is more to it. Among the elements which need to be
explained are:
- Euclid’s avoidance of numbers in describing lengths, areas, and so on.
- His use—notably in book II—of a geometric language for manipulating areas where his
Egyptian or Babylonian predecessors would have used something more like what we call
algebra—see the discussion of proposition II.1 in the Introduction. - His theory of ‘ratios’ which are intended to replace numbers in contexts where these may not
be fractions (area of circle to square on diameter, for example, see below).
Clearly, the most economical hypotheses to explain all this would be: first, that Euclid’s practices
were the result of a ‘second mathematical revolution’; and second, that this second revolution
was—on Kuhn’s model—the result of serious problems which arose in the original practice of
rigorous mathematics which made it impossible to proceed. Both of these theses, originally put
forward around 1910, are still widely believed in a revised form; we now need to examine the
arguments for and against them.
Exercise 5.How would you prove Thales’ statement on isosceles triangles, and what assumptions would
you need?
7. Drowning in the sea of Non-identity
“But betray me,” said Neary, “and you go the way of Hippasos.”
“The Akousmatic, I presume,” said Wylie. “His retribution slips my mind.”
“Drowned in a puddle,” said Neary, “for having divulged the incommensurability of side and diagonal.”
“So perish all babblers,” said Wylie.
“And the construction of the regular dodeca—hic—dodecahedron,” said Neary.” (Beckett 1963, p. 36)
- Nevertheless, you can of course still find long discussions of what Pythagoras did, on the Internet and even in ‘general’ histories.