There is a curious connection between the combinatorics of finite sets and projective geometry over a finite field. Namely, we can consider a finite set as a vector space over a field with one element - but no such field exists! Recently much fuss has been made over this field with one element, but the literature is quite technical. In this talk I will discuss some elementary issues related to the analogy between combinatorics and finite projective geometry, including the way in which Dynkin diagrams represent simultaneously the symmetries of finite sets and of projective spaces. The one-element field remains a mystery, but with it as my muse I hope to be able to bring to light some surprising connections between seemingly disparate areas of mathematics.
The talk will be completely elementary.