Incomputability after Alan Turing

Those of us old enough may remember being fascinated by George Gamow’s popular books on mathematics and science—with the most famous being One Two Three...Infinity. Gamow got us to imagine living on the surface of a two-dimensional balloon with only two-dimensional experience of the surface. And then he got us to understand how we might detect its three-dimensional curved character via purely two-dimensional observations. In Figure 1 is his picture from page 103 of the 1961 edition. Algorithms, as a way of traversing our four dimensions, have been with us for literally thousands of years. They provide recipes for the control and understanding of every aspect of everyday life. Nowadays, they appear as computer programs. Algorithms, or computer programs, can be thought of as a kind of causal dimension all their own. Then the questions arise: Is there a causal dimension that is not algorithmic? Does it matter if there is? Notice that Gamow’s example showed that on one hand it was tricky to live in two dimensions and find evidence of a third. But it did matter precisely because we could find that evidence. Of course, if we took the mathematical model presented by the picture, the missing dimension becomes clear to us—we have an overview. But observe that while the mathematical overview gives us a better understanding of the nature of curved space, it does not tell us that the model is relevant to our world. We still need to look at the triangle from within the two-dimensional world to match up reality and mathematics and to be able to apply the full power of the model. Back in the 1930s, people such as Kurt Gödel, Stephen Kleene, Alonzo Church, and Alan Turing did build mathematical models of the computable dimension of causal relations. This enabled Church and Turing to get outside this dimension and to use their models to explore the new dimension of incomputability. A specially important part of what the twentyfour-year-old Alan Turing did was to base his investigation of the extent of the computable on a new machine-like model. The Turing machine was to make him famous in a way no one could have foreseen. He had the idea of using Gödel’s coding trick to turn Turing machine programs into data S. Barry Cooper is professor of mathematics at the University of Leeds and chair of the Turing Centenary Advisory Committee. His email address is pmt6sbc@maths. leeds.ac.uk.