[OPE] A dialectical view of mathematics...

The assumption of the paper seems to be that the development of mathematical
ideas can be fully explained in terms of a logic internal to the discipline.
But this assumes that mathematics does not have an effect on the rest of the
social world, and that the social world has no effect on mathematical
activity. This seems hardly credible, and so, to explain mathematical
development (why, for example, some problems are pursued while others are
ignored) one would have to refer both to endogenous influences (internal to
the discipline) and exogenous ones (external to it). In that case, the
dialectic is not simply a dialectic of ideas, but also an interaction
between ideas and practical life circumstances, and a dialectic of societal
development. If the development of mathematical ideas can be explained fully
in terms of a logic internal to the discipline, we should be able to make
useful predictions about the actual future trajectory of mathematical
development based on its current structure, but in reality the predictions
have often failed rather spectacularly, in the sense that they did not
anticipate new developments that were also logical possibilities implied by
existing knowledge.

Jurriaan

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Received on Sun Feb 22 18:17:46 2009