I'll tell the dramatic story of the recent disputes over the foundations of mathematics. I'll start with the problems in Cantor's theory of infinite sets and then discuss the work of Bertrand Russell, David Hilbert, Kurt Godel and Alan Turing, and finally my own work using complexity. This complexity-based analysis of the foundations of mathematics suggests to me that perhaps mathematics is more similar to physics and to biology than is commonly believed, and should sometimes be carried out quasi-empirically, that is, more in the spirit of an experimental science.