44 A History ofMathematics
or dogmatic assertion. To be more cautious, to insist as Lloyd does on how little we know, is to risk
being found unexciting.
To some extent the status of the revolution in mathematics is linked to the others, particularly
philosophy (given Plato’s views on mathematics as a model for reasoning). From a different
viewpoint, though, mathematics could gain by being considered separately. In the first place,
the innovations in mathematics were of a very specific nature and have led to quite particular
accounts of the revolution; and in the second, to see mathematics as the product of ageneral
revolutionary process makes it too homogeneous.6. Two revolutions?
[The Greeks] were certainly not the first to develop a complex mathematics—only the first to use, and then also to give
a formal analysis of, a concept of rigorous mathematical demonstration. (Lloyd 1979, p. 232)
Greek mathematical deduction was shaped by two tools: the lettered diagram and the mathematical language. (Netz,
1999, p. 89)So far, I have concentrated on the most obvious novelty in Greek mathematics—the use of an
ordered sequence of deductions, which in Euclid appears to be from what is considered self-evident.
This interpretation of Euclid is strengthened by Aristotle’s prescriptions (for mathematics and
deductive science in general), for which see Fauvel and Gray 2.H.1, p. 93–4. It is this revolution
which is of interest to Lloyd, partly for the reason that his study concerns the wider field of rational
discourse in Greek society, whether medical, philosophical, or mathematical; and his description of
what was distinctive to the Greeks in the first quote is a good one. In the second quote Netz, looking
at the collection of documents we have as evidence of how a community worked throughout the
Greek period, concurs in seeing what was distinctive about Greek mathematics as a unified practice
based on a ‘toolkit’ of argument about lettered diagrams. However, any search for the origin of this
particular revolution, apart from general sociological speculation of the type outlined above, takes
us back to the most unreliable parts of the commentators’ story—to Thales (early sixth century
bce) and Pythagoras (around 500bce). Proclus claims, for example, that ‘old Thales’ proved six
results, one being that the base angles of an isosceles triangle are equal (1970, p. 250/195). In
other words, if the triangle ABC has sides AB, AC equal then so are the angles B, C.
This would certainly be ‘revolutionary’ if we had any reliable evidence (which, as far as Thales
is concerned, Proclus is not). One should not discount the value of myth and late propaganda as
historical source material. But it is, in the main, source material for ‘what later Greeks thought
about their origins’ rather than for ‘what Thales did’. The idea that such a fact, which Thales’
predecessors could well have considered obvious, needed proving, and the attempt, whatever it
might be, to construct a proof, would have marked Thales’ geometry off from any earlier ideas of
what geometry (traditionally, the measurement of land...) was about.
Speculation about what Thales did, which was once an acceptable part of Greek mathematical
history, is now generally discounted as serious history (except as a metaphor, perhaps, for example,
by Michel Serres, 1995, p. 105). The same is true, if anything more so, for Pythagoras, who
Proclus claimed founded ‘pure’ mathematics (‘transformed mathematical philosophy into a scheme
of liberal education’, 65/53). The scholarship of Walter Burkert in particular (see 1972), has
established fairly conclusively that no mathematical discoveries can be soundly attributed to