He was a visionary in artificial intelligence at the time when the first modern computers were developed around John von Neumann. During the war, working on the logic of algorithms, he broke the encryption of German Enigma machines previously considered as inviolable.
On April 13 in New York, Bonhams sells a highly rare notebook, lot 1. The scientific thoughts by Turing, written after 1942, are laid at the start and end of the book, while the intermediate pages were used for personal notes by his eminent student Robin Gandy after the ignominious death of Turing (who was pardoned posthumously in 2013 by Queen Elizabeth II).
Both autographs by Turing are unpublished developments. They bring a new vision on the thinking process of one of the most profound scientists of our time, who found through a method of enforcing simplifications the solutions of problems that his illustrious predecessors had just managed to define.
The first of them is a review of Peano's axioms concerning the completeness and consistency of mathematics. After Hilbert, Turing also addresses the third problem of Peano on decidability, meaning the possibility of establishing an algorithm to decide whether a mathematical proposition is true or false. This text is anticipating his involvement with Enigma but certainly helps to understand the thinking process that enabled the deciphering.
The second scientific text of the book is a study of the role of the mathematical notation. Turing appreciated that a complex formulation makes it more difficult to find the solution. His approach is based on the search for a universal language of mathematics by Leibniz. Such researches make him a precursor of computer programming.