> In the discussion so far, there are two rather different kinds of questions.
> One is the general question about the extent to which some form of cultural
> relativism exists in the mathematical approaches of different cultures, and
> the other is about the extent to which the Greeks found some way of doing
> 'mathematics' which was in some way superior to that of other cultures. The
> latter point is often associated with the use by the Greeks of an approach
> to mathematics which is described as axiomatic.
[snip]
It seems to me that the great and unique contribution of classical Greek
mathematics is not specifically the axiomatic method but, more generally,
the idea of proof. To return to the starting point of this discussion--my
guess is that the affinity that Hardy and Littlewood felt with classical
Greek mathematics was not so much to do with axiomatization (their own work
was certainly quite remote from this!) but with the more general notions of
proof and rigour.
Of course, the idea that mathematical propositions can and *should* be
established by conclusive argument, by proof, may--if taken to its ultimate
conclusion--lead to axiomatization. But the achievements of classical Greek
mathematics (think, for example, of Archimedes) can by no means be reduced
to axiomatics.
To sharpen the point: Archimedes may well have borrowed or inherited some
algorithms from ancient Egyptian mathematics, as Milo Gardner has claimed.
But the point is that Archimedes must have recognized the need, and had the
ability, to validate these algorithms by a general proof. Is there any
evidence that the Egyptian priests who invented these algorithms were able
to supply such proofs or even to see any need for them? I doubt it.
In this sense, Greek mathematics represents enormous progress. By this I do
not intend to belittle the achievement of earlier mathematicians, possibly
in other civilizations, which made that progress possible.
Whether this progress is a *good* thing is an entirely different matter.
For example, you may belong to a culture in which the idea of proof is
considered very bad and dangerous, as it tends to undermine people's trust
in the authority of the priestly elite, and thus contribute to social
unrest. And, moreover, proofs may be a waste of time if you are mainly
interested in basket weaving.
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