Mathematics has always been a necessary component in modern warfare. During the World War II era, mathematicians Alan Turing and John von Neumann were responsible for some of the technological and scientific developments which contributed Allied victory. After considering their accomplishments before the war, their contributions during the war, and how they were recognized after the war, you will see that each mathematician is remembered very differently for their contributions. Turing is barely honored for his code breaking techniques, which helped preemptively end the war with the use of nonviolent programmable machinery, while von Neumann is honored and respected for ending the war by developing nuclear weapons, something that many people fear to this day.
Turing produced some noteworthy achievements leading up to his involvement in World War II. Born June 23, 1912, Alan Mathison Turing was recognized by his preliminary and secondary school teachers for having a natural talent in the subjects of mathematics and science, while having mediocre talents in the non-sciences. In 1931, inspired by the death of a childhood friend who also had exemplary skills in the sciences, he decided to receive his undergraduate studies in Mathematics at the University of Cambridge's King's College, UK(Dyson, 459). In 1935, he became a fellow of King's University after completing a dissertation on the Central Limit Theorem which showcased his mathematical genius(Dyson, 459). That same year, his interest in solving David Hilbert's "decision problem" led to his paper, "On computable numbers, with an application to the Entscheidungsproblem."
Hilbert's problem questions whether a mechanical procedure can determine the truth of any logical statement in a finite number of iterations. Turing's solution is heavily influenced by the work of Kurt Gödel’s Incompleteness Theorem which states, "no consistent system of axioms whose theorems can be listed by an effective algorithm is capable of determining the truth about relations of the natural numbers"(Wikipedia). In order to solve the problem, Turing conceived the idea of the Turing machine: a black box that that obeys preset instructions which are encoded as symbols stored in the internal state of mind of the machine(Dyson, 459). His response to the Hilbert's open problem was no, and proved that it would take an uncountable number of steps to determine the truth of any given logical statement using such machines. John von Neumann paraphrased Turing's solution in a lecture by saying, "you can build an organ which can do anything that can be done, but you cannot build an organ which tells you whether it can be done"(Dyson, 460). Turing's work essentially pioneered the programming of mathematical logic into machines, and was published in 1936 while he was graduate student at Princeton University, New Jersey(Dyson, 460). By 1936 he completed his PhD and returned to Britain at the outbreak of World War II.