In 1939 at University of Cambridge Wittgenstein lectures on the nature of the foundations of mathematics and logic. Wittgenstein's work on the philosophy of mathematics was of interest to several mathematically trained students who decide to attend Wittgenstein's lectures. In particular, Donald C. MacPhail establishes a relationship with Wittgenstein, becoming the "star pupil" in Wittgenstein's 1939 lectures.

MacPhail, stimulated by Wittgenstein's iconoclastic views of logic, devises a "calculus of contradiction". This new branch of mathematical logic is basically a method for working with a large system of theorems, some of which are contradictory. MacPhail and Wittgenstein realize that MacPhail's methods for dealing with mathematical contradiction are applicable to the semantic networks that form inside human brains when children learn. MacPhail applies Markov chains in the "calculus of contradiction" and publishes a paper on the subject with Wittgenstein: "Population Dynamics of Logical Atom Fields Containing Contradictions". This article attracts the attention of Turing, eventually leading to constructive interactions between Turing and Wittgenstein during World War II. Ultimately, this leads to profound changes in the development of machine learning and autonomous robots with significant progress towards human-like artificial intelligence.