Chapter 3. Mathematical Cultures II
Many cultures borrow from others but add their own ideas to what they borrow and
make it into something richer than the pure item would have been. The Greeks, for
example, never concealed their admiration for the Egyptians or the debt that they
owed to them, yet the mathematics that they passed on to the world was vastly
different from what they learned in Egypt. To be sure, much of it was created in
Egypt, even though it was written in Greek. The Muslim culture that flourished
from 800 to 1500 CE learned something from the Greeks and Hindus, but also
made many innovations in algebra, geometry, and number theory. The Western
Europeans are still another example. Having learned about algebra and number
theory from the Byzantine Empire and the Muslims, they went on to produce such a
huge quantity of first-rate mathematics that for a long time European scholars were
tempted to think of the rest of the world as merely a footnote to their own work.
For example, in his history of western philosophy (1945), the British philosopher
Bertrand Russell wrote (p. xvi), "In the Eastern Empire, Greek civilization, in a
desiccated form, survived, as in a museum, till the fall of Constantinople in 1453, but
nothing of importance to the world came out of Constantinople except an artistic
tradition and Justinian's Codes of Roman law." He wrote further (p. 427), "Arabic
philosophy is not important as original thought. Men like Avicenna and Averroes
are essentially commentators." Yet Russell was not consciously a chauvinist. In
the same book (p. 400), he wrote, "I think that, if we are to feel at home in the
world after the present war [World War II], we shall have to admit Asia to equality
in our thoughts, not only politically, but culturally."
Until very recently, many textbooks regarded as authoritative were written
from this "it all started with the Greeks" point of view. As Chapter 2 has shown,
however, there was mathematics before the Greeks, and the Greeks learned it before
they began making their own remarkable innovations in it.
1. Greek and Roman mathematics
The Greeks of the Hellenic period (to the end of the fourth century BCE) traced
the origins of their mathematical knowledge to Egypt and the Middle East. This
knowledge probably came in "applied" form in connection with commerce and
astronomy/astrology. The evidence of Mesopotamian numerical methods shows
up most clearly in the later Hellenistic work on astronomy by Hipparchus (second
century BCE) and Ptolemy (second century CE). Earlier astronomical models by
Eudoxus (fourth century BCE) and Apollonius (third century BCE) were more
geometrical. Jones (1991, p. 445) notes that "the astronomy that the Hellenistic
Greeks received from the hands of the Babylonians was by then more a skill than
a science: the quality of the predictions was proverbial, but in all likelihood the
practitioners knew little or nothing of the origins of their schemes in theory and
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