  Incomputability as a mathematical notion arose from work of Alan Turing and
Alonzo Church in the 1930s. Like Turing himself, it attracted less attention
than it deserved beyond the confines of mathematics. Today our experiences in
computer science, physics, biology, artificial intelligence, economics and the
humanities point to the importance of the notion for understanding the world
around us. This article takes a Turing centenary look at how the interface
between the computable world and the incomputable formed a central theme in
Turing's work - from the early establishment of the standard model of the
stored program computer, to the late work on emergence of form in nature, and
the first approaches to understanding and simulating human intelligence.
Turing's thinking was remarkably prescient, and his legacy still impacts on
much of our work. The incomputable may turn out to be a specially important
part of the legacy, with consequences that Alan Turing himself could not have
envisaged.
