The Legend of Abraham Wald
Historians of science sometimes lament the fact certain historical figures are famous for the wrong reasons. Alan Turing (1912-54) is perhaps the best example of this phenomenon. Most people associate Turing with the breaking of an encryption device called the German Navy Enigma, rather than his—arguably—much more important contributions to theoretical computer science. Another example is Albert Einstein (1879-1955), who is typically recognized for his mass-energy equivalence formula E = mc² rather than for general relativity. J. Robert Oppenheimer (1904-67) is another one, more frequently associated with being the director of the Manhattan Project than with the Born-Oppenheimer approximation, arguably a much more important scientific contribution.
In this essay I will argue that Abraham Wald (1902-50) is another such figure. His name, for those who recognize it, is today more or less synonymous with the term survivorship bias (the selection bias of focusing on those observations which have survived past some selection process, overlooking those which have not). In particular, Wald is remembered for his inclusion of survivorship bias in calculations attempting to describe how to minimize bomber plane losses during World War II . The graphical depiction of his work (see image above) summarizes his finding:
Reinforce the areas of the planes which are unscathed. The planes hit in those areas are the most likely to be lost—as they haven’t returned to be included in the analysis of damages.
Although clever, this is far from Wald’s most important or even impressive contribution to science.
This essay is about Abraham Wald, the brilliant mathematician whose insights into geometry would eventually spawn a revolution in economics.
Early Life (1902-26)
Abraham Wald was born on October 31, 1902 in the city of Kolozsvár, Transylvania in what was then the Kingdom of Hungary, modern-day Romania. The city has since changed its name to Cluj-Napoca. His father was an orthodox Jewish business man and his grandfather a famous rabbi. He had five siblings. His older brother Martin was an electrical engineer and inventor.
Although intellectually gifted, Wald could not be admitted at the local gymnasium because he, as the son of an orthodox Jew, was unable to attend school on Saturdays as his family observed the Shabbat (Wolfowitz, 1952). Instead he was home-schooled until passing the “matura” (A-level exams), qualifying him for entrance at the King Ferdinand I University in Cluj, now Babeș-Bolyai University.
The University of Vienna (1927-31)
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 theory and 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).
In February of 1930, Wald returned to Vienna. A soon as he arrived, he reached out to Menger to “hand [him] a manuscript which purported to contain the solution of a famous problem. It was a serious piece of work, but an error at the very end invalidated the result.” Upon hearing the news, as Menger wrote,
“Wald was visibly disappointed. But a few days later he returned to tell me that, during the last week, he had been sitting in on my lectures on metric geometry—the first university lectures he ever attended—and that he planned to follow this entire course. Moreover, he wanted to try his hand at some problem in this field.”
Menger challenged Wald to come up with a characterization of “betweenness” among ternary relations in a metric space:
The point q is between the points p and r if, and only if, , p ≠ q ≠ r and the three distances between the points satisfy the equality:
d(p, q) + d(q, r) = d(p, r)
Wald brought Menger a first draft four weeks later, asking for a new problem. The characterization he came up with was subsequently published in the prestigious Mathematische Annalen, one of Wald’s first scientific publications (Menger, 1952).
Wald, A. 1931. Axiomatik des Zwischenbegriffes in metrischen Räumen (“The Axiomatics of Betweenness in Metric Spaces”). Math. Ann. 103, pp. 476-484.
The Lévy-Steinitz theorem (1933)
At some point in 1933, Wald grew interested in the Lévy-Steinitz theorem, named after mathematicians Paul Lévy (1886-1971) and Ernst Steinitz (1871-1928). Lévy had in 1905 identified the set of values to which rearrangements of an infinite series of vectors in Rⁿ can converge. In 1913, Steinitz filled a gap in Lévy’s proof, as well as also proving the result by a different method. The theorem may be described in the following way (Rosenthal, 1987):
The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translation of a subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace).
The result was a generalization of Bernhard Riemann (1826-1866)’s finding that any not absolutely converging series of real numbers can, by a permutation of its terms, be made to converge toward any number. The Lévy-Steinitz theorem states that the vectors of an n-dimensional space (toward which a series of vectors can, by a permutation of terms, be made to converge) form a linear manifold. Wald provided a new (third) proof of the theorem and extended it to spaces of infinitely many dimensions (Menger, 1952). His results were published in two papers in 1933:
Wald, A.1933. Vereinfachter Beweis des Steinitzschen Satzes über Vektorenreihen im Rₙ (“A Simplified Proof of Steinitz’ theorem about Vector Series in Rₙ“). Ergebnisse eines Math. Kolloquiums 5, pp. 10-13.
Wald, A. 1933. Bedingt konvergente Reihen von Vektoren im Rω (“Conditionally Convergent Series of Vectors in Rω“). Ergebnisse eines Math. Kolloquiums 5, pp. 13-14.
Lebesgue measure (1935)
Wald’s supervisor Menger’s work in geometry in the 1920s included the formulation of postulates for the concept of dimensions. In order to enhance the analogy between these postulates and those for the Lebesgue measure (developed by Henri Lebesgue, 1875-1941), Wald developed a characterization of the Lebesgue measure among set functions in which he confined the additivity postulates to closed sets (Menger, 1952).
Wald, A. 1935. Eine Charakterisierung des Lebesgueschen Masses (“A Characterization of Lebesgue’s Measure“). Ergebnisse eines Math. Kolloquiums 6, pp. 27-29.
On the Curvature of Surfaces (1935-36)
His supervisor Menger in 1952 described that Wald wrote his “masterpiece in the field of pure mathematics” on the topic of curvature. He wrote:
“By virtue of the triangle inequality d(p, q) + d(q, r) ≥ d(p, r) three points p, q, r of a metric space are always congruent to three points of the Euclidean plane.
Consider the circum-circle of these latter points and call the reciprocal of its radius the curvature of the points p, q, r. This curvature is zero if and only if one of the three points is between the other two. Now let A be an arc contained in a metric space. Its points need not be given by coordinates, and its shape is not necessarily described by equations or functions. All that is assumed is an ordered continuum with a distance defined for every pair of point.”
Menger had defined the curvature of A at the point a as the number (if it exists) from which the curvature of any three points differ by arbitrarily little, if and only if all three points are sufficiently close to the point a (Menger, 1952). Although numerous results had been proven about this general curvature of curves—including modifications by Menger’s student Franz Alt (1910-2011) and Hahn’s student Kurt Gödel (1906-78)—one problem remained: the extension of the same idea to higher-dimensional manifolds. Menger described the state of the problem as follows:
“From the outset it had been clear that, on a surface, quadruples of points should be considered. But what number should be associated with a given quadruple of points of a metric space? Four congruent points in the euclidean space do not necessarily exist; and even if they do exist, the radius of their circum-sphere is of no particular significance".
Wald tackled the problem by considering spheres metrized by the length of the arcs of great circles. “For positive k, let Sₖ denote the sphere of curvature k thus metrized; S₀ is the Euclidean plane; for negative k, let Sₖ be the hyperbolic plane of curvature k. If four points of a metric system are given, what Sₖ contains all four congruent points?” Wald proved that if S is a surface of the type studied in classical differential geometry, then for each point a there exists a number κ(a) with the following property: in S, for every quadruple of points each of which is sufficiently close to a, there exists a congruent quadruple in an Sₖ of which the curvature k differs arbitrarily little from κ(a). Moreover, he proved that the number κ(a) is equal to the Gauss curvature of S at the point a (Menger, 1952). As Menger wrote, “even if he had stopped at this point, his result would have been a remarkable achievement. But here was the beginning of Wald's really great work”.
In the second part of Wald’s 1936 paper, he dropped the assumption that a surface of the type studied in classical differential geometry be given. That is, he dispensed with the characterization of points by coordinates and of surfaces by equations, functions or parametrization. Extending Menger’s idea, he instead assumed a compact metric space S with the following properties (Menger, 1952):
S is convex; and
S at every point a has curvature κ(a);
By convexity is meant that for any two points p and r of S, there exists a point between p and r. By curvature κ(a) is meant that for any four points of S which are sufficiently close to a, there exists four congruent points on a sphere Sₖ where k differs arbitrarily little from κ(a). From these simple assumptions, Wald could deduce the following of S:
S is a surface;
In this surface, polar coordinates can be locally introduced;
In terms of these coordinates, the length is expressed as it is on the classical surface of differential geometry; and
For each a of S, the number κ(a) is equal to the Gauss curvature at a of the classical surface created on S by the introduction of the polar coordinates;
Wald’s theorem not only addressed the lacking definition of curvature for higher-dimensional manifolds. With his two simple assumptions, he also showed “serious redundancies” in the existing assumptions used in classical differential geometry (Menger, 1952).
Wald, A. 1935. Zur Differentialgeometrie der Flächen: I. Eine neue Definition der Flächenkrümmung (“On the Differential Geometry of Surfaces: A New Definition of Surface Curvature“). Ergebnisse eines Math. Kolloquiums 6, pp. 29-39.
Wald, A. 1935. Sur la courbure des surfaces (“On the Curvature of Surfaces“). C. R. Acad. Sci. Paris 201, pp. 918-920.
Wald, A. 1936. Begründung einer koordinatenlosen Differentialgeometrie der Flächen (“Justification for a Coordinate-Free Differential Geometry of Surfaces“). Ergebnisse eines Math. Kolloquiums 7, pp. 24-46.
Between 1931 and 1937 Wald published 21 papers in geometry, an incredibly impressive output for any academic—even more so for one who was only able to work on his research on nights and weekends. As Menger wrote, “When he ceased working in the field of geometry, it was not for lack of interest. It was for lack of time”.
Menger’s Vienna Colloquium (1928-37)
Wald’s advisor Menger had in 1927 been invited by his Ph.D. supervisor Hans Hahn (1879-1934) to attend the famous Vienna Circle, at that point chaired by Moritz Schlick (1882-1946). As Menger later wrote
“A remarkable feature of Vienna in those years was the existence of numerous Kreise (circles, in the sense of discussion groups), some with, some without, direct ties with the academic world. […] I knew about these circles since most of them included one or another of my friends; but personally I did not belong to any circle, except one that was non-scientific. Until one day in the fall of 1927, Hahn said to me: ‘I hope you can join our philosophical circle around Schlick. We meet informally about every other Thursday evening in Schlick’s Institute.”
Students in Menger’s department later asked their professor to arrange a seperate colloquium dedicated specifically to mathematical questions such as those Wald was working on, a mathematisches kolloquium:
“In these meetings, topics and results in their [students’] and my fields of interest were reported and discussed. We followed the unconstrained style of the Schlick Circle; but from the fall of 1929 on I kept a Protokollbuch — minutes of a sort — something which Schlick, as far as I know, unfortunately never did.”
Among the first non-students of Menger at his colloquium was Gödel, who at that point was a Ph.D. student working on under Hahn. He first attended at the end of 1929 (and went on to graduate in 1930). According to Menger:
“From then on [Gödel] was a regular participant, not missing a single meeting when he was in Vienna and in good health. From the beginning, he appeared to enjoy these gatherings and poke even outside of them with members of the group. […] He was a spirited participant in discussions on a large variety of topics. Orally, as well as in writing, he always expressed himself with the greatest precision and at the same tim with the utmost brevity. In nonmathematical conversations he was very withdrawn.”
Other attendees at the time included Georg Nöbeling (1907-2008), Alt and Olga Taussky-Todd (1906-95). Wald first attended the colloquium at Menger’s invitation as he (Menger) believed that “Wald had exactly the spirit which prevailed among the young mathematicians who gathered together […] so I at once invited him to present his result [on betweenness] there.” He did so in May of 1930. Read more about Menger’s Vienna Colloquium and its remarkable list of attendees here:
Despite Wald’s performance both as a student and postgraduate, he was not able to obtain a university position in Vienna after graduating. This, primarily for two reasons. Firstly, in general, there were very few open academic positions in Vienna at the time. However, had there been, such positions would not have been available to Jews, no matter how brilliant. As an Ostjude (eastern Jew) and member of the poor immigrant community that “flooded” Vienna after World War I, offering a position to Wald was simply outside of the question for any serious academic institution. As Düppe and Weintraub (2015) wrote:
“The immigrants from the eastern parts of the former Austro-Hungarian Empire created a number of problems for interwar Vienna. Vienna’s intellectual elite looked down upon them. The established Viennese Jewish community, with such luminaries as Sigmund Freud, Otto Neurath, and Hans Mayer were uncomfortable with their mostly poor co-religionists.”
Wald’s accent and appearance thus made life in Vienna difficult for him, in a time of ongoing political strife between liberals and nationalist parties—a time of economic crisis and increasing overt anti-Semitism (Düppe and Weintraub, 2015). As Menger wrote after Wald’s death, “it was out of the question to secure for him a position at the University of Vienna, although such a connection should certainly have been as profitable for that institution as for himself”. Thus, Wald was forced to return to Cluj. He did so in 1931, but remained in correspondence with Menger via letter, sending him new ideas, expositions and proofs regularly.
Back in Cluj, Wald was still unable to find suitable work. By the summer of 1932 he was writing to Menger that “my prospects of getting a position are not favourable […] I would extremely happy if it would be possible for me to do scientific studies for a year in Vienna.” With characteristic modesty, despite his brilliance Wald told Menger that he would be perfectly satisfied with any small private position which would enable him to continue his work in Menger’s colloquium.
Menger remembered that his friend Karl Schlesinger (1889-1938), a Viennese banker and economist had expressed wishes to broaden his knowledge of higher mathematics. Menger knew Schlesinger from connections to his fathers’ economists’ circle, the Nationalökonomische Gesellschaft (the Austrian Economics Society). He recommended that if Schlesinger was interested, Wald could tutor him, a proposition both men duly accepted. Wald was thus able to return to Vienna in December of 1932.
Another person in the market for tutoring lessons in mathematics in Vienna at the time was economist Oskar Morgenstern (1902-77). At that point, Morgenstern was a privatdozent of political economics at the University of Vienna, as well as the co-director of the Austrian Institute of Economic Research (WIFO) alongside Friedrich Hayek (1899-1992). In 1931, Morgenstern assumed sole directorship at the institute. As he was interested in mathematics, under Morgenstern’s directorship WIFO went on to sponsor a series of lectures on the “Mathematics of Economists”, to be given by Menger with help from his students Wald and Alt (Sigmund, 2017). In 1933, Menger approached Morgenstern to offer Wald a position at WIFO. As Morgenstern later wrote, “Menger, aware of the quite exceptional gifts and great mathematical power of his pupil, tried to find money wherever possible and he approached me in 1933 because, as the director of the Austrian Institute for Business Cycle Research and with and interest in applications of mathematics to economics and statistics, I might find some funds”.
“So Wald and I became acquainted and we soon developed an intimate friendship that lasted to his death. Like everyone else, I was captivated by his great ability, his gentleness, and the extraordinary strength with which he attacked his problems” — Oskar Morgenstern, 1951
And so, differential geometer and doctor of mathematics Wald was hired both as a statistician at Morgenstern’s institute and as his personal tutor of mathematics (succeeding Menger’s other student Franz Alt). As far as can be gathered, he had never taken a single course on statistics in university.
Mathematical Economics (1933-37)
Schlesinger was the first attendee at Menger’s colloquium to present work relevant to economics, in 1934. Up until that point, on account of Menger and his students’ interests, geometry and topology had figured most prominently (Düppe & Weintraub, 2015). Schlesinger’s paper was titled Über die Produktionsgleichungen der ökonomischen Wertlehre (“On the Production Equations of Economic Value Theory“) and was presented orally on the 19th of March 1934. In it, Schlesinger suggested a modification of Léon Walras (1834-1910)’ set of equations describing a general equilibrium, which he (Walras) had never proved to have solutions except arguing that they should have based on the number of equations and unknowns (Debreu, 1998). For this and other reasons, the ‘Walrasian model’ had fallen by the wayside until it was resurrected by Swedish economist Gustav Cassel (1866-1945) when he used it as inspiration for his 1918' treatise Theoretische Sozialökonomie (“The Theory of Social Economy”), translated to English in 1923.
Schlesinger’s insight was to point out that the supply of free commodities (of which demand is smaller than supply) should not be assumed have a number a priori, but rather that their supply (like all other supply) should be determined by the system of equations of general equilibrium, each free commodity having a price of zero (Debreu, 1998). This observation and modification of Walras and Cassel’s theory would later turn out to be essential. His (seemingly) trivial insight inspired Wald (who was attending the colloquium where he presented his claims) to, that same day, prove the existence of the general equilibrium which had evaded Walras decades prior.
His first attempt, given somewhat strict conditions was presented on March 19th. He soon after also presented a proof given much weaker conditions, on November 6th. His papers—one using a model based on a system of exchange and the other based on a model of production and exchange—were published in the proceedings of the colloquium in 1934:
Wald, A. (1934a). Über die eindeutige positive Lösbarkeit der neuen Produktionsgleichungen, ("On the Unique Non-Negative Solvability of the New Production Functions"). Ergebnisse eines mathematischen Kolloquiums, 1933-34.
Wald, A. (1934b). Über die Produktionsgleischungen der ökonomischen Wertlehre (2. Mitteilung), (“On the Production Equations of Economic Value Theory (Part 2)“). Ergebnisse eines mathematischen Kolloquiums, 1934-35.
Wald’s two papers signify an important moment in the history of mathematical economics. Although the existence of the Walrasian equilibrium had been proposed as early as in the 1870s, the mathematical tools available to Walras never permitted him to show that such solutions must exist, given reasonable assumptions. Although the assumptions of Wald’s model are not very reasonable according to modern standards (including the assumption that there was only one consumer in the market), he was the first to solve the equilibrium problem that had plagued economists up until that point (Düppe & Weintraub, 2014).
"Accordingly, to a follower of Menger, the determination of economic equilibrium would not merely involve the determination of the prices of those goods which have prices (as in Walras); it should also involve the determination of which goods are to have prices and which are to be free. The weakness of the Walras-Cassel lies in the implied assumption that the whole amount of each available factor is utilized; once that assumption is dropped, the awkwardness of the construction....can be shown to disappear." — John Hicks, 1960
According to Morgenstern Wald had little awareness of the significance of his findings for economics. To him, he was working on mathematical problems with mathematical solutions which had limited mathematical implications. Indeed, Morgenstern had to ‘badger’ Wald to write a summary of his two papers, which were later published for an audience of economists in the prestigious Zeitschrift für Nationalökonomie (where Morgenstern served as editor from 1930-38) in 1936. The paper was also later translated to English by Otto Eckstein and published in the prestigious journal Econometrica:
Wald, A. 1936. Über einige Gleichungssysteme der mathematischen Ökonomie(“On some systems of Equations in Mathematical Economics”), Zeitschrift für Nationalökonomie 7(5). p. 637-670.
The assumptions underlying Wald’s proofs would later be greatly improved by economists Kenneth Arrow (1921-2017), Gérard Debreu (1921-2004) and Lionel W. McKenzie (1919-2010), one of Morgenstern’s later Ph.D. students at Princeton University. Their extensions now constitute the foundation of modern general equilibrium theory, the so-called Arrow-Debreu model.
By early 1935, Morgenstern was writing the Rockefeller Foundation—which had provided a grant for Wald’s position—praising his statistical and mathematical work, noting that there was “still very much purifying to be done” within mathematical economics (Morgenstern, 1935):
“On the basis of the experiences of the last years I have worked out a program for research which I beg to outline briefly. […] I am absolutely convinced that abstract theoretical work, even making use of mathematical analysis or of the modern methods of Logic that have not yet been applied to Economics, are just as necessary as the systematic collection of facts.”
Wald’s would continue working in mathematical economics in Vienna for another two years, publishing an additional five papers (in parallel with his work and considerable output in differential geometry):
Wald, A. 1936. Über die produktionsgleichungen der ökonomischen Wertlehre (Mitteilung II) (“On the Production Equations of the Economic Theory of Value“) Ergebnisse eines Math. Kolloquiums 7, pp. 1-6.
Wald, A. 1936. Über einige Gleichungssysteme der mathematischen Ökonomie (“On Some System of Equations in Mathematical Economics“). Zeitschrift für Nationalökonomie 7, pp. 637-670.
Wald, A. 1937. Zur Theorie der Preisindexziffern (“The Theory of Price Index Numbers“). Zeitschrift für Nationalökonomie 8, pp. 179-219.
Wald, A. 1937. Grundsätzliches zur Berechnung des Produktionsindex (“Principles for Calculating the Production Index“). Monatsberichte des Österreichischen Institutes für Konjunkturforschung 11 (Appendix 6), pp. i-vii.
Wald, A. 1937. Extrapolation des gleitenden 12-Monatsdurchschnittes (“Extrapolation of the 12-Month Moving Average“). Monatsberichte des Österreichischen Institutes für Konjunkturforschung 11 (Appendix 8), pp. i-vii.
“[Abraham] Wald is really smart. I think these [new] works are very important; at the same time, they shed new light on the application of mathematics in economics. Without them, you won’t be able to survive at all now. Hayek is a donkey.”— Oskar Morgenstern, 1935
In 1937, Wald received an invitation from businessman Alfred Cowles (1891-1984) to come to Colorado Springs to join the staff of the Cowles Commission as a statistician. In addition to his work in geometry and mathematical economics, Wald had also published on statistics and probability after joining WIFO. He was however not eager to leave Vienna—despite the fact that tensions against Jews in Austria were at an all time high.
In January of 1933 the President of Germany, Paul von Hindenburg (1847-1934), reluctantly appointed Nazi leader Adolf Hitler (1889-1945) as the new Chancellor of Germany. After only two months in office, following the burning of the Reichstag building, the German parliament passed the "Enabling Act" giving the chancellor full legislative power for a period of four years. Exploiting the death of von Hindenburg the next year, Hitler used the act to merge the offices of Chancellor and President, creating the new office of "Führer and Reichskanzler".
In Vienna, a governmental crisis in 1932 had given way for a conservative politician named Engelbert Dollfuss (1892–1934) to ascend to power in Austria. In early 1933, responding to Hitler's own ascension to Chancellor of Germany, Dollfuss shut down the Austrian parliament and assumed dictatorial power. The next February a brief Austrian Civil War—sometimes known as the February Uprising—took place when austrofascists loyal to Dollfuss skirmished with socialist forces. By July, taking advantage of the unstable situation, Austrian Nazis loyal to Hitler had stormed the Chancellery in a failed coup and assassinated Dollfuss. In response, the head of the Italian fascist party Benito Mussolini (1883-1945) mobilized part of the Italian army on the Austrian border, threatening Hitler with war in the event of a German invasion of Austria. With Mussolini's approval, Dollfuss was soon succeeded by Kurt Schuschnigg (1897–1977) who restored stability in Austria.
By 1938, pressure from pro-unification activists had forced the Austrian chancellor Schuschnigg to announce that there would be a referendum on the 13th of March on a possible union with Nazi Germany. Hitler portrayed the vote as an "obfuscation of the popular will in Austria and Germany" and threatened invasion. Under the pressure, Schuschnigg cancelled the referendum. Unopposed by the Austrian military, on March 12th the German Wehrmacht marched into Austria in what is now known as the Anschluss, the annexation of Austria into Nazi Germany.
The proceedings of Menger’s colloquium, the Ergebnisse, had by the time of the Anschluss come under scrutiny—with specific reference to Wald—for its large number of Jewish contributors. As Menger later wrote,
“Hahn was dead. Schlick had been assassinated. Viennese culture resembled a bed of delicate flowers to which its owner refused soil and light while a fiendish neighbour was waiting for a chance to ruin the entire garden. I left the country. A year later Hitler marched into Vienna. Schlesinger, who occupied a rather prominent position, chose death that same day.”
Wald stayed in Vienna for a few weeks after the invasion. Morgenstern had been in Princeton when Hitler invaded and so decided to remain in the United States. His successor at WIFO, “the new Nazi director” (Morgenstern, 1951) dismissed Wald. Morgenstern and Menger both tried to convince him to leave for America immediately. Menger went to the University of Notre Dame in 1937.
Wald eventually did travel to the United States—via his native Romania—in 1938. According to Morgenstern, Wald “took quickly to America. He was cheered by the friendliness with which he was received and stimulated by the genuine interest in his work and plans”. He first joined the Cowles Commission as a staff member, but soon received an invitation from Harold Hotelling (1895-1973) to instead come work as a research associate at Columbia as a Fellow of the Carnegie Corporation. He left for New York in the fall of 1938 and was within a year elected a Fellow of the American Statistical Association. He was appointed to the faculty of Columbia University in 1941 as an assistant professor and made a professor of mathematical statistics three years later, in 1944.
Life in America (1938-50)
Wald’s sisters, brother Martin, as well as their spouses and children nearly all—all but one of nine family members, his brother Hermann—died in Auschwitz crematoria during World War II.
After emigrating to the United States in 1938, Wald met a woman named Lucille Lang whom he married in 1941. The couple had two children, Robert Wald (1947-) and Betty. In 1950, Wald and his wife both travelled to India on invitation from the Indian government for Wald to give a lecture tour. While travelling from Calcutta to Bangalore, the Air India plane they were travelling in crashed in the Nilgiri Mountaints, killing everyone on board. Wald was 48 years old.
“Abraham Wald was taken from the scientific community and his many personal friends at the height of his life. He had already given more than almost anyone in his fields and was full of new ideas that awaited development. These are lost to us, but the work he finished will live to guide and instruct more than one generation to come.”— Oskar Morgenstern, 1951
Following his death, his friends and supporters Menger and Morgenstern both published papers discussing Wald’s life and accomplishments, as did economist Georg Tintner (1907-83) and statistician Jacob Wolfowitz (1910-81). These are available below:
Morgenstern, O. 1951. Abraham Wald, 1902-1950. Econometrica 19(4), pp. 361-367.
Menger, K. 1952. The Formative Years of Abraham Wald and his Work in Geometry. The Annals of Mathematical Studies 23(1), pp. 13-20.
Tintner, G. 1952. Abraham Wald’s Contributions to Econometrics. The Annals of Mathematical Statistics 23(1), pp. 21-28.
Wolfowitz, J. 1952. Abraham Wald, 1902-1950. Annals of Mathematical Statistics 23(1), pp. 1-13.
Wald published more than sixty books and papers from 1938 to his death, more than one hundred in his lifetime. A full list of his publications is available here.
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Casselman, B. 2016. The Legend of Abraham Wald. American Mathematical Society Feature Column.
Debreu, G. 1998. Foreword: Economics in a Mathematics Colloquium. In: Dierker, E., Sigmund, K. (eds). Karl Menger. Springer, Vienna.
Düppe, T. & Weintraub, E.R. 2015. Losing Equilibrium. On the Existence of Abraham Wald’s Fixed-Point Proof of 1935. CHOPE Working Paper No. 2015-04.
Menger, K. 1952. The Formative Years of Abraham Wald and his Work in Geometry. The Annals of Mathematical Studies 23(1), pp. 13-20.
Morgenstern, O. 1951. Abraham Wald, 1902-1950. Econometrica 19(4), pp. 361-367.
Sigmund, K. 2017. Exact Thinking in Demented Times: The Vienna Circle and the Epic Quest for the Foundations of Science*. Basic Books.
Weintraub, E.R. (1992). Towards a History of Game Theory*. Duke University Press
Wolfowitz, J. 1952. Abraham Wald, 1902-1950. Ann. Math. Statist. 23(1), pp. 1-13.
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** The title of this essay was stolen from Bill Casselman’s column on the American Mathematical Society (AMS)’s website. Read his wonderful essay. (I’m sorry Bill, it’s just too good not to nick).