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.

If you're interested in this, you might be interested in David Corfield's page about the Philosophy of Real Mathematics. "Real" doesn't mean "as opposed to complex" here - it roughly means "what real mathematicians actually do". He has written a very interesting book entitled "Towards a Philosophy of Real Mathematics".

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.