> I would like to ask a heretical question.
>
> Is there no *objective* progress, no development, no advance in
> mathematics? Would we not be justified in thinking that our present
> mathematics (as practiced and tought in universities around the world) is
> *objectively* much more advanced than--and in *this* sense superior to--the
> mathematics of the ancient world? And along this *objective* scale of
> progress, is not the mathematics of ancient Greece (as exemplified, say, in
> the writings of Archimedes) considerably more advanced than the mathematics
> of ancient Egypt and Mesopotamia, as far as we can tell from the documents
> that have come down to us? Or are such non-relativist thoughts simply the
> result of class bias, Eurocentrism and the damage inflicted by imperialism
> on our thinking?
There is a big difference between not knowing the answer to a mathematical
question one day, and discovering it the next. In this sense, the situation
today (knowing) is better than the situation was yesterday (not knowing).
There is a definite temporal asymmetry which we can interpret as "mathematical
progress". This time-directional phenomenon is genuine: once a result enters
the literature (the collective consciousness), we can be confident that this
new knowledge will not become wrong at some time in the future (modulo errors
that may be discovered in these results, of course). This stands in sharp
contrast to the situation in other disciplines (philosophy, art, politics,
etc.).
On the other hand, the "social" aspects of mathematics are real. Much new
mathematics is forgotten and is rediscovered later, or is discarded as no
longer interesting and falls out of the collective consciousness. (What
happened to all the work done in determinantal identities at the end of the
last century and beginning of this? and what is now happening to synthetic
geometry?) How do we evaluate this as progress?
Danny Otero