I think that exposing the difficulties and failures throws much light
on the concepts involved. The idea that one should view the history of
mathematics as a history of "successes" purveys the false view that
the growth of mathematics is a linear process, with a very clear idea
of what "progress" means.
We know that "progress" is not clear in a new field. For example, just
remember the many different definitions (of course all equivalent) of a
topological vector space which strove for supremacy at the beginnings of
point set topology, till the definition of a topology via a distinguished
collection of subsets, called open subsets, imposed itself on the other
equivalent definitions.
And what about the whole theory of "invariants" that Hilbert disposed of?
I think that the history of mathematics should focus on both the false or
mistaken roads taken as well as on the brilliant and deep insights. Both
are numerous and should be considered.
Joao Bosco Pitombeira
Mathematics Department
PUC-Rio
Rio de Janeiro
Brazil