Greeks and‘Origins’ 41
of mathematicians scattered among other writers. While Plato and Aristotle date from the
fourth-centurybce(and are in different ways very important sources), the rest are over 500 years
later, and are commentators with particular philosophical allegiances. They have often been
treated, as they should be, with suspicion, but for much of the early history they are all we
have. A particularly important source is Proclus, who, around 450ce, wrote a commentary
on Euclid’s book I, prefaced by a general history of mathematics up to Euclid’s time and an
exposition of what, in general, the aim of theElementswas (the most recent edition is Proclus
1970). For a number of statements, Proclus is the only source; but besides his late date, his
allegiance to Plato’s philosophy and to later embroideries of it make him a source to be treated
with caution.
Third, all the works which exist are not—like the works of the Babylonians or Egyptians—
‘contemporary documents’ (manuscripts of Archimedes from his time or near it, say). As in ancient
Egypt, papyrus was used for writing, and it is extremely perishable. Manuscripts were copied and
recopied over centuries, and the earliest surviving copy of Euclid’sElementsis from the ninth-
centuryce, over 1000 years after the work was written. (Details on the earliest manuscripts are
given in Fowler 1999, p. 218ff) Worse, mathematicians see no need to respect a text they are
copying if they can explain it or express its meaning more simply, and for some works the earliest
manuscripts have certainly been edited in this way. This problem is not as serious as the others, but
it exists.
More interestingly, we know very little about what kind of people in the ancient Greek world did
mathematics, and why. Archimedes wrote a few interesting letters to patrons and colleagues, but
he gives no explanation of why mathematicians are engaged in their solitary activity; and, as Netz
points out (1999, p. 284–5), the community he refers to is extremely small. And this, one would
think, was at the high point of Greek mathematics. Most of the time, the mathematicians seem
not to have left records describing their practice, their students, their aims, and thoughts about
their work. In this respect, our situation is worse than for ancient Egypt, say, where mathematics,
so far as we know, was done by a socially well-defined group of people for particular reasons. A
useful start at ‘profiling’ the world of Greek mathematicians is done by Netz (1999, ch. 7), but it is
necessarily speculative.^5
In general, if we want a good basis for writing the history of any activity (witchcraft in the
seventeenth century, say, or the Vietnam War), we would like to have documents which describe
what the participants were doing; how contemporary observers saw it; and the general social
setting in which the events took place. What we have for Greek mathematics is nothing like this.
Rather, it consists of an impressive collection of major texts—less than 20 in number—with some
more minor ones, and some late and unreliable stories. The material spans a period of 1000 years
or more. If we want to think about it in a historical way, how do we do so? More contentiously, one
could ask: why do we want to?
Without even attempting the second question, we should make a particular point about
‘reconstruction’, which is one of the few methods available for dealing with the first. Whenever
the historical record is weak, one wants to fill in the gaps, supplementing slender materials with
imagination so as to form an idea of what Boudicca’s chariots were like, what was in the Holy Grail,
or the identity of Jack the Ripper. In mathematics, we attempt a reconstruction of a particular
form: briefly, to deduce an unknown proof from the fact that we are told that one existed. This can
- In particular, Netz’s conclusion that the number of Greek mathematicians over the whole recorded history was at most
1000 gives the classic works, and in particular theElements, a tiny readership; unless one distinguishes a higher-level ‘creative’
mathematician from someone educated well-enough to read the basic works.