Turing and von Neumann’s Brains and their Computers Dedicated to Alan Turing’s 100 th birthday and John von Neumann’s 110 th birthday Sorin Istrail* and Solomon Marcus** In this paper we discuss the lives and works of Alan Turing and John von Neumann that intertwined and inspired each other, focusing on their work on the brain. Our aim is to comment and to situate historically and conceptually an unfinished research program of John von Neumann, namely, towards the unification of discrete and continuous mathematics via a concept of thermodynamic error; he wanted a new information and computation theory for biological systems, especially the brain. Turing’s work contains parallels to this program as well. We try to take into account the level of knowledge at the time these works were conceived while also taking into account developments after von Neumann’s death. Our paper is a call for the continuation of von Neumann’s research program, to metaphorically put meat, or more decisively, muscle, on the skeleton of biological systems theory of today. In the historical context, an evolutionary trajectory of theories from Leibniz, Boole, Bohr and Turing to Shannon, McCulloghPitts, Wiener and von Neumann powered the emergence of the new Information Paradigm. As both Turing and von Neumann were interested in automata, and with their herculean zest for the hardest problems there are, they were mesmerized by one in particular: the brain. Von Neumann confessed: “In trying to understand the function of the automata and the general principles governing them, we selected for prompt action the most complicated object under the sun – literally.” Turing’s research was done in the context of the important achievements in logic: formalism, logicism, intuitionism, constructivism, Hilbert’s formal systems, S.C. Kleene’s recursive functions and Kurt Gödel’s incompleteness theorem. Turing’s machine, exclusively built on the paper, as an abstract computing device, has been the preliminary theoretical step towards the programmable electronic computer. Turing’s 1937 seminal paper, one of the most important papers in computer science, prepared the way for von Neumann’s 1948 programmable computer. Von Neumann’s unfinished research program was outlined in his seminal articles “The general and logical theory of automata” (1951) and “Probabilistic logics and the synthesis of reliable organisms from unreliable components” (1956), his posthumous book The Computer and the Brain (1958) and the unfinished book The Theory of SelfReproducing Automata, completed and published by A. Burks (1966). He proved in 1948, inspired by Turing’s universal machine, part of his theory of selfreproduction of automata, five years before Watson and Crick, the structure of the DNA copying mechanism for biological self reproduction. Biologist and Nobel Laureate Sydney Brenner, in his memoirs, acknowledges von Neumann’s prophetic theorem: “You would certainly say that Watson and Crick depended on von Neumann, because von Neumann essentially tells you how it’s done.” 1. The Duo Were it not for two decades of the intertwined intellectual lives of Alan Turing and John von Neumann the disciplines of mathematics and computer science would not be what they are today.