Turing Minimalism and Complexity

Not only did Turing help found one of the most exciting areas of modern science (computer science), but it may be that his contribution to our understanding of our physical reality is greater than we had hitherto supposed. Here I explore the path that Alan Turing would have certainly liked to follow, that of complexity science, which was launched in the wake of his seminal work on computability and structure formation. In particular, I will explain how the theory of algorithmic probability based on Turing’s universal machine can also explain how structure emerges at the most basic level, hence reconnecting two of Turing’s most cherished topics: computation and pattern formation Alan Turing established such a direct connection between the concept of the algorithm and a purely mechanical process that he left few doubts concerning the physical implementation and generality of his programmable machines. At the beginning of the twentieth century and through the end of the Second World War, computers were human, not electronic, mainly women. The work of a computer consisted precisely in solving tedious arithmetical operations with paper and pencil. This was looked upon as work of an inferior order. At an international mathematics conference in 1928, David Hilbert and Wilhelm Ackermann suggested the possibility that a mechanical process could be devised that was capable of proving all mathematical assertions. This notion is referred to as Entscheidungsproblem (in German), or ‘the decision problem’. If a human computer did no more than execute a mechanical process, it was not difficult to imagine that arithmetic would be amenable to a similar sort of mechanization. The origin of Entscheidungsproblem dates back to Gottfried Leibniz, who having (around 1672) succeeded in building a machine based on the ideas of Blaise Pascal that was capable of performing arithmetical operations (named Staffelwalze or the Step Reckoner), imagined a machine of the same kind that would be capable of manipulating symbols to determine the truth value of mathematical principles. To this end Leibniz devoted himself to conceiving a formal universal language, which he designated characteristica universalis, a language which would encompass, among other things, binary language and the definition of binary arithmetic. In 1931, Kurt Godel arrived at the conclusion that Hilbert’s intention (also referred to as ‘Hilbert’s programme’) of proving all theorems by mechanizing mathematics was not possible under certain reasonable assumptions. Godel advanced a formula that codified an arithmetical truth in arithmetical terms and that could not be proved without arriving at a contradiction. Even worse, it implied that there was no set of axioms that contained arithmetic free of true formulae that could not be proved. In 1944, Emil Post, another key figure in the development of the concepts of computation and computability (focusing especially on the limits of computation) found that this problem was intimately related to one of the twenty-three problems (the tenth) that Hilbert, speaking at the Sorbonne in Paris, had declared the most important challenges for twentiethcentury mathematics. Usually, Hilbert’s programme is considered a failure, though in fact it is anything but. Even though it is true that Godel debunked the notion that what was true could be proved, presenting a negative solution to the ‘problem of decision’, and Martin Davis (independently of Julia Robinson) used Godel’s negative result to provide a negative solution to Hilbert’s tenth problem (the argument for which was completed by Yuri Matiyasevich), Hilbert’s supposedly