Argument and debate form cornerstones of civilized society and of intellectual life.  Processes of argumentation run our governments, structure scientific endeavour and frame religious belief.  As online interaction usurps many traditional forms of interaction and communication, we would hope to see these processes alive and well on the web.  But we do not.  Current mechanisms for online interaction hamper and discourage debate; they facilitate poor quality argument; and they allow fuzzy thinking to go unchecked.  Meanwhile, these same  online resources are increasingly being trusted and adopted with little critical reflection.  To address the problem, we need new tools, new systems and new standards engineered into the heart of the internet to encourage debate, to facilitate good argument, and to promote a new online critical literacy.  This talk extends my inaugural presentation at the Discovery Days 2010, exploring both the tools and the mathematics behind them, reviewing the computational uses to which theories of argument are put.