This is the title of a talk I gave on 23rd January, 2004, at the
Cambridge University Society for the Philosophy of Mathematics.
This is an excellent seminar run by students and attended by both
philosophers and mathematicians of all levels, from undergraduate up to
professor. The format is: a 45 minute talk followed by 45 minutes
of invariably lively discussion (which usually spills into the
cafeteria afterwards).

Abstract:

A source of tension between Philosophers of Mathematics and
Mathematicians
is the fact that each group feels ignored by the other; daily
mathematical
practice seems barely affected by the questions the Philosophers are
considering. In this talk I will describe an issue that does have an
impact on mathematical practice, and a philosophical stance on
mathematics that is detectable in the work of practising mathematicians.

No doubt controversially, I will call this issue 'morality', but the
term
is not of my coining: there are mathematicians across the world who use
the word 'morally' to great effect in private, and I propose that there
should be a public theory of what they mean by this. The issue
arises because proofs, despite being revered as the backbone of
mathematical truth, often contribute very little to a mathematician's
understanding. 'Moral' considerations, however, contribute a great
deal. I will first describe what these 'moral'
considerations might be, and why mathematicians have appropriated the
word
'morality' for this notion. However, not all mathematicians are
concerned
with such notions, and I will give a characterisation of 'moralist'
mathematics
and 'moralist' mathematicians, and discuss the development of
'morality'
in individuals and in mathematics as a whole. Finally, I will
propose
a theory for standardising or universalising a system of mathematical
morality,
and discuss how this might help in the development of good mathematics.

I have written this up and incorporated my slides into the text.
This
is available in pdf: click here.

You might also be interested in the fascinating webpages of John Baez. He is a
truly great expositor on all sorts of things including mathematics and
mathematical physics. His comments on Corfield's book can be
found here.