implicitly) that *there is* an objective progress in mathematics.
He claims that classical Greek mathematics has not been *shown* to be more
advanced than ancient Egyptian and Babylonian mathematics. But I note that
he does not go so far as to claim that, according to the available
evidence, Greek matheamtics can be shown to be *not* more advanced than
ancient Egyptian and Babylonian mathematics.
He does not answer the question as to whether our present mathematics is
much more advanced than ancient mathematics. I believe that, on the
available evidence, the answer must be a definite `yes'.
Be that as it may, I think that historians, who display such exemplary
caution in making value judgements about the mathematics of various
civilizations, should not rush to condemn Hardy and Littlwood as
class-biased snobs, under the ideological influence of imperialist
thinking. This seems to me a case of some histrians reading the
Hardy/Littlewood dictum in the light of their (the historians') own
pre-conceived sociological-ideological bias, without due consideration of
the evidence.
++++++++++++++++++++++++++++++++++++++
At 3:50 pm -0600 4/11/98, Glen Van Brummelen wrote:
>I have thought about this question in my paper. As you say, our
>present mathematics contains immeasurably more content than any
>non-Western culture. In fact, it contains the entire content of these
>ancient results (as we know them) as a tiny subset, and often more
>precise arguments to support them.
>
>But:
>
>-- Comparison of the *amount* of mathematics produced confuses the
>*actual* power/superiority with *potential* power. Western
>mathematics has had immeasurably more resources devoted to it, and
>there is no way to answer whether, say, the medieval Chinese approach
>would have produced more/better results, given the equivalent
>centuries and personnel applied to it.
This seems to me extremely plausible. Humans in other civilizations did not
differ from us in ingenuity or other relevant attributes.
Glen goes on:
>-- Continuing my fantasy, I imagine that a modern, well-developed
>"Chinese mathematics" would develop in ways quite distinct from our
>own. Our desire to reduce to axioms is not present in Chinese
>mathematics, whereas some of their (fantasized modern) computational
>math might be well beyond ours. We value the study of analysis and
>transcendental numbers, Lebesgue integration, and so forth at least
>partly because of the relations that arise when we axiomatize (and
>study pathological cases); the Chinese might prefer to emphasize
>other subdisciplines.
I find this also very plausible. Progress need not be uni-directional.
>Who is to say which emphasis is better? We can make a firm conclusion
>only by agreeing at the outset on what we value about mathematics,
>and that is culturally bound. (That being said, a culturally bound
>answer is not then automatically valueless; just restricted).
I refrained from using value terms such as `better' (in the sense used
here). *More advanced* is clearly not at all the same thing as *better*.
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