Following his undergraduate studies in mathematics, Wald applied to (but was barred from entering) the University of Vienna for graduate studies. The university had quotas for Jewish students. Nonetheless, Wald still travelled to Vienna. He entered an engineering school for a year before reaching out to the mathematical institute at the University of Vienna in 1927. As mathematician Karl Menger (1902-85) later wrote:
“In the fall of 1927, a man of 25 called at the Mathematical Institute of the University of Vienna. Since he expressed a predilection for geometry he was referred to me. He introduced himself as Abraham Wald. In fluent German, but with an unmistakable Hungarian accent, Wald explained that he had carried on most of his studies at the elementary and secondary school levels at home, mainly under the direction of his older brother Martin, a capable electrical engineer […] He had just arrived in Vienna in order to study mathematics at the university. Geometry had interested him ever since he was fourteen.”
– Excerpt, The Formative Years of Abraham Wald and his Work in Geometry by Karl Menger (1952)
Wald shared with Menger, by then professor of geometry, that he had been reading David Hilbert (1862-1943)’s Grundlagen der Geometrie (Foundations of Geometry) and saw possibilities for improving Hilbert’s work by “omitting some postulates and weakening others”. Menger suggested to Wald that he write up his results, which Wald did. The paper was later published in the third volume of Ergebnisse eines Mathematischen Kolloquium, the proceedings of Menger’s mathematics colloquium (read: Karl Menger’s Vienna Colloquium, 1928-36). Wald’s proofs, one of which was later incorporated in the seventh edition of Hilbert’s book, were included in the paper:
- Wald, A. 1931. Über das Hilbertsche Axiomensystem der Geometrie (“On Hilbert’s Axioms of Geometry”) Ergebnisse eines Math. Kolloquiums 3, pp. 23-24.
Although Menger was sufficiently impressed to ensure that Wald could enrol for graduate studies studies in his department, the professor would not see much of his student as Wald shortly thereafter was called to serve in the Romanian army. However, as Menger wrote, “the system of complete freedom which at the time prevailed in the universities of Central Europe kept the gifted students from wasting semesters on courses the content of which they could absorb in a few weeks of concentrated reading”. Thus, Wald was still able to keep up with his studies while in the army. In the four years that he studied under Menger he only attended three courses, on metric geometry, dimension theoryand lattice operations. Wald graduated in 1931 with a Ph.D. in mathematics. His thesis was entitled Über das Hilbert’sche Axiomensystem (“On Hilbert’s System of Axioms“) (Düppe & Weintraub, 2015) and dealt with a question of axiomatics (Wolfowitz, 1952).