42 A History ofMathematics
be illuminating; but, unless a document which confirms the reconstruction turns up, it must
necessarily be provisional, leaving the field open for other competitors.5. The Greek miracle
The king moreover (so they say) divided the country among all the Egyptians by giving each an equal square parcel of
land...And any man who was robbed by the river of a part of his land would come to Sesostris and declare what had
befallen him; then the king would send men to look into it and measure the space by which the land was diminished, so
that thereafter it should pay in proportion to the tax originally imposed. From this, to my thinking, the Greeks learned
the art of geometry...(Herodotus, cited in Fauvel and Gray 1.D.4 (a), p. 21)
What the Greeks discovered — the greatest discovery made by man — is the power of reason. It was the Greeks of
the classical period, which was at its height during the years from 600 to 300 B.C., who recognized that man has an
intellect, a mind which, aided occasionally by observation or experimentation, can discover truths. (Kline (1980),
p. 9.)The problem of why Greek mathematicians wrote their works as they did tends to be framed
as a problem of origins: there must have been some event, some discovery perhaps, which led to
what is called the ‘Greek revolution’ or ‘Greek miracle’ in mathematics. The quotation from Morris
Kline is a typical example. There is a case for using revolutionary language, on the analogy of the
sixteenth century ‘scientific revolution’; and one might try to adopt a version of Kuhn’s theory of
scientific revolutions. An obvious difficulty is our ignorance of when or how it might have taken
place. The Greek authors are not responsible for the idea that they were revolutionary, but that
should not stop us from adopting it if it seems reasonable. The ‘paradigm’ (to use Kuhn’s term) of
doing mathematics for the known Greek authors is a different one from any which went before. The
evidence that we have, suggests that their predecessors and indeed contemporaries in Egypt and
Mesopotamia were better at calculation, but quite unconcerned with formal proof.^6 However, if we
accept that there was a revolution, the study of its origin has been a very problematic one, and it
does not seem well adapted to Kuhn’s framework in which the idea of a ‘scientific community’ who
pursue normal science is an essential component. We have a good record of what could be called a
scientific community in the ancient Near East, even if it may not have been much like our own, but
our knowledge of such a community at the dawn of Greek mathematics is almost non-existent.
Before the 1920s, and the publication of the Egyptian and Babylonian texts, it was universally
believed that the Greek revolution actually founded mathematics as a science—and hence, the first
truly scientific discourse. The belief lingers on among those who, either because they wish to define
scientificmathematics in such a narrow way that it excludes the sophisticated procedures of the
Babylonians, or because they have an ideological investment in a Western origin, cannot accept
the evidence that Greek mathematicians, however revolutionary, weretransformingan earlier
Middle Eastern practice. To take one example, an important French ‘structuralist’ tradition adopts
the hard science/prescience distinction made by Gaston Bachelard in the 1930s (the so-called
‘epistemological break’ which founds a science, see Bachelard 2003). In the 1960s, Louis Althusser- One should always here remember how very fragmentary our sources are. Both Herodotus (see the quote at the opening of
this section) and Aristotle described mathematics as an Egyptian import (see Fauvel and Gray extracts 1.D.4), while Plato in hisLaws
(extract 2.E.6), probably for propagandist reasons, claimed that the teaching of mathematics was better in Egypt than in Athens.
However, the state of mathematics in Egyptat that time(fifth–fourth centurybce) is unknown to us.