## Richard Von Mises

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## Von Mises, Richard

# Von Mises, Richard

Richard von Mises (1883-1953), who contributed notably to the field of applied mathematics, was born in Lemberg, in the Austro-Hungarian Empire. His father, Arthur von Mises, held a doctoral degree from the Institute of Technology in Zurich and was a prominent railroad engineer in the civil service. On his travels all over the empire he was often accompanied by his family, and von Mises was born on one of these journeys. The family home was in Vienna. Von Mises was the second of three brothers; the eldest, Ludwig, is an economist of international reputation; the youngest died while still a boy. His father’s family included engineers, physicians, bankers, and civil servants. Among the members of his mother’s family were philologists and bibliophiles.

Von Mises attended the Akademische Gymnasium in Vienna and graduated in 1901 with high distinction in Latin and mathematics. He then studied mechanical engineering at the Vienna Technical University. In 1906, immediately after finishing these studies, he became an assistant to Georg Hamel, who had just accepted a professorship of mechanics at the Technical University in Brunn (now Brno). In 1908 von Mises was awarded a doctorate by the Technical University in Vienna and in the same year obtained the *venia legendi (Privatdozentur)* at Brunn, with an inaugural dissertation entitled *Theorie der Wasserrader* (1908). But after only one year (at the age of 26), he was called to Strasbourg as associate professor of applied mathematics, the field he made famous.

After five happy and fruitful years in Strasbourg, von Mises joined the newly formed Flying Corps of the Austro-Hungarian Army at the outbreak of World War I (he already had a pilot’s license). He was soon recalled from service in the field to act as technical adviser, organizer, and instructor. In the *Fliegerarsenal* in Aspern, he taught the theory of flight to German and Austrian officers; these lectures constituted the first version of his *Fluglehre* (1918), which went through many editions. He was commissioned to design the first large airplane of the empire, the “Grossflugzeug.” At the same time he was working on two basic papers on probability (discussed below).

When the war was over, von Mises could not return to Strasbourg, which had become French. After a brief interlude as lecturer in Frankfurt, he was called in 1919 to the Technical University in Dresden as professor, and in 1920 to the University of Berlin as professor of applied mathematics and director of the Institute of Applied Mathematics. This institute was actually founded by him and was a precursor of several similar institutes in Europe and America. In 1921 he founded the *Zeitschrift filr angewandte Mathematik und Me-chanik,* the first journal of its kind. As its editor until 1933, he exerted a profound influence on applied mathematics all over the world. For von Mises, applied mathematics included mechanics, practical analysis, probability and statistics, and some aspects of geometry and philosophy of science. He educated a generation of young applied mathematicians. His first assistant was Hilda Geiringer, who held a PH.D. in “pure” mathematics but turned, under his influence, to applied mathematics. She became his collaborator and later his wife.

When, in 1933, von Mises recognized that it would be both unwise and undignified to remain in Berlin, he accepted the position of professor of mathematics and director of the mathematical institute in Istanbul, Turkey, at the university that had been revitalized by Kemal Atatiirk. He reorgan-ized the institute, lectured in French and in Turkish, maintained close relations with Turkish professors and dignitaries, and became a leading figure at the university. But in 1939, with the approach of World War n, he felt he had to leave Istanbul; and he accepted a position as lecturer in the School of Engineering at Harvard University. There, he was appointed, in rapid succession, associate professor and Gordon McKay professor of aerodynamics and applied mathematics. He continued his own scientific work as well as the education of undergraduates, postgraduates, and research workers.

The fields to which von Mises made distinctive contributions are (1) mechanics and geometry, (2) probability and statistics, (3) philosophy of science, and (4) analysis. Of these, the first two categories occupied him the most. Geometry captivated him all his life, and most of his geometric contributions are closely connected with mechanics.

The outstanding feature of his work is a striving for clarity and complete understanding. In his contributions to mechanics no vague statements, no *ad hoc* engineering theories are tolerated; explanations of observations follow strictly from the principles of mechanics. Particularly important achievements are his *Theorie der Wasserrader* (1908); his wing theory (1917-1920), which is based on con-formal mapping; and his celebrated work on plasticity (1913; 1925; 1928a; 1949).

The main directions of von Mises’ thought on the theory of probability appeared in his first major papers on the subject, the “Fundamentalsatze der Wahrscheinlichkeitsrechnung” and the “Grundlagen der Wahrscheinlichkeitsrechnung,” both of 1919. Von Mises considered probability as a science of the same epistemological type as, say, mechanics. Its mathematical construction is distilled from experience. The main concept, introduced in the “Grundlagen,” is the *Kollektiv* (also denoted as “irregular collective"), which, in the simplest case, idealizes the sequence of results of the repeated tossings of a coin under unaltered circumstances. The collective as a mathematical notion is thus an infinite sequence of zeros and ones (heads and tails). If among the first *N* terms of the sequence there are *N0* zeros and N1 ones, *N0 +* Nt = *N,* the *frequencies N0/N, N,/N* are given. For reasons of mathematical expediency it is then assumed that, in the abstract sequence, the *limits of these frequencies* exist as *N* tends toward infinity. In addition, the infinite sequence is to have the property of *randomness;* vaguely explained, this means the following: if we consider not all *N* trials (not all *N* terms of the sequence) but only the second, fourth, sixth, … or only those whose number is a prime number or only those which follow a run of three “ones,” we obtain by such a *selection* a frequency *N[/N’* (and *N’0/N’),* and it is postulated that for any such selection

This p_{1} is the *probability* of the result one, and p_{0} = 1 — P_{1} is that of the result zero. Randomness is the mathematical equivalent of the “impossibility of a gambling system” and thus characterizes the sequences which form the subject of probability calculus.

Von Mises then built up probability theory, by means of collectives, in one or more dimensions. In 1938 Abraham Wald proved the “consistency”— i.e., existence in the mathematical sense—of the collective, indicating precise conditions. Von Mises accepted Wald’s results as a necessary and valuable complement. He felt that in mathematics, as well as in any other science, the unceasing improvement and refinement of existing concepts must parallel the creation and extension of new concepts.

Von Mises’ theory is in contrast with the a prior-istic theory of Laplace, whose definition of probability is both logically unsatisfactory and too narrow. Laplace and his followers therefore had to distinguish between a “theoretical” and an “empirical” probability; the mathematical theorems proved with the theoretical definitions were then unhesitatingly applied to problems where Laplace’s “equally likely” and “favorable” events often failed to exist. Von Mises showed in a penetrating analysis that for modern probability, as used in physics, biology, and some of the social sciences, Laplace’s definition is quite insufficient.

Von Mises’ frequency theory also differs from today’s abstract measure-theoretical approach, most closely associated with Kolmogorov. The contrast is not between “frequency” and “measure": in von Mises’ developments, as well as in Kolmo-gorov’s, both frequencies and measures are essential. Von Mises wanted to lay the conceptual foundations of the science of probability; Kolmogorov, the axiomatic foundations of the calculus of probability. [See Probability.]

The “Fundamentälsatze” deals with two basic general problems. (1) Given *n* distributions (for example, *η* dice with given probabilities *p _{i}^{(v)} i*= 1,2, …, 6;

*v =*1,2,…,

*n*for the six faces of the dice), with results X

_{v}in the

*vth*trial, what is the distribution, as

*n*→∞, of the sum X

_{1},…,X

_{n}(equivalently, of the average) of these results? Regarding this group of problems, indicated here by a very special case, von Mises proved in 1919 two basic “local” theorems and studied the most general problem. (2) Perhaps an even more important contribution is his formulation and study of the second fundamental problem. Consider again a very special case. A coin with

*unknown*heads-probability is thrown

*η*times, and “heads” turn up Hi times. What inference can we make from this observed result about the unknown heads-probability of the coin? Obviously, this is the typical problem of inference from an observed

*sample*to an unknown “theoretical” value. This problem was considered by von Mises as the crucial problem of theoretical statistics. This “Bayesian” point of view has been widely attacked by R. A. Fisher and his students but seems to be more and more accepted today. For von Mises, statistics was just one (very important and general) application of probability theory.

In the last years of his life von Mises introduced the fundamental concept of a *statistical function* (as important as the concepts earlier introduced by him of *distribution* and of *sample space),* which led to vast generalizations of the two problems of the “Fundamentalsatze.” Von Mises’ work in probability and statistics is incorporated in many papers and in three books: his *Wahrscheinlichkeitsrech-nung* of 1931 (Volume 1 of *Vorlesungen aus dem Gebiete der angewandten Mathematik),* a comprehensive textbook of his theory; his *Probability, Statistics and Truth* (1928b), a lucid presentation in nontechnical language of his foundations of probability and their applications in statistics, biology, and physics; and his lecture notes, *Mathematical Theory of Probability and Statistics* (1964), which restates and extends the foundations of the theory and builds on them a unified theory of probability and statistics, with particularly original contributions to statistics.

Von Mises did not believe that statistical explanations in physics—and other domains of knowledge—are of transient utility while deterministic theories are the definite goal. He thought that a judgment of what constitutes an “explanation” is, like anything else, subject to change and development. The “Laplacean daimon” of complete deter-minacy is no longer accepted, nor is an immutable law of causality. Philosophers, von Mises thought, are apt to try to “eternalize” the current state of scientific affairs, just as Kant held Euclidean space as an absolute category. In contrast with these “school philosophers,” he called himself a “posi-tivist.” In an address given shortly before his death he said, “He is a positivist who, when confronted by any problem reacts in the manner in which a typical contemporary scientist deals with his problems of research.” Von Mises thought of science in the general sense of the German *Wissenschaft.* In his book *Positivism* (1939) he followed up this conception through the various domains of thought and of life.

Von Mises loved poetry: He could recite long passages from Goethe, as well as from such modern poets as Hofmannsthal, Verlaine, Altenberg, and, in particular, Rilke. In Rilke’s esoteric poetry he found a confirmation of his belief that in areas of life not yet explored by science, poetry expresses the experiences of the mind:

Nicht sind die Leiden erkannt, nicht ist die Liebe gelernt, und was im Tod uns entfernt,

ist nicht entschleiert. Einzig das Lied iiberm Land heiligt und feiert.

Pain we misunderstand,

love we have yet to learn,

and death, from which we turn,

awaits unveiling.

Song alone circles the land

hallowing and hailing.

Sonnets to Orpheus,First Part, xix.Frankfurt am Main: Insel, 1923.

London: Hogarth, 1936.

Von Mises was a recognized authority on the life and work of Rilke. Over a lifetime, he compiled the largest privately owned Rilke collection (now at Harvard’s Hough ton Library), for which a 400-page catalogue was published in 1966 by the Insel Verlag, Leipzig.

Hilda geiringer

*[For the historical context of von Mises’ work, see the biography of*laplace; *for discussion of the subsequent development of von Mises’ ideas, see the biographies of*fisher, R. A.; Wald.]

## WORKS BY VON MISES

1908 *Theorie der Wasserrädder.* Leipzig: Teubner. → Reprinted from Volume 57 of the *Zeitschrift filr Mathematik und Physik.* Partially reprinted in Volume 1 of von Mises’ *Selected Papers.*

1913 Mechanik der festen Körper im plastisch-deforma-blen Zustand. Gesellschaft der Wissenschaften, Got-tingen, Mathematisch-Physikalische Klasse *Nach-richten*[19131:582-592.

1917-1920 Zur Theorie des Tragflächenauftriebs. *Zeitschrift filr Flugtechnik und Motorluftschiffahrt* 8:157-163; 11:68-73, 87-89.

(1918) 1957 *Fluglehre: Theorie und Berechnung der Flugzeuge in elementarer Darstellung.* 6th ed. Edited by Kurt Hohenemser. Berlin: Springer.

1919a Fundämentalsatze der Wahrscheinlichkeitsrech-nung. *Mathematische Zeitschrift* 4:1-97.

1919b Grundlagen der Wahrscheinlichkeitsrechnung. *Mathematische Zeitschrift* 5:52-100.

1925 Bemerkungen zur Formulierung des mathematischen Problems der Plastizitatstheorie. *Zeitschrift fur angewandte Mathematik und Mechanik* 5:147-149.

1928a Mechanik der plastischen Formänderung von Kristallen. *Zeitschrift fur angewandte Mathematik und Mechanik* 8:161-185.

(1928fc) 1957 *Probability, Statistics and Truth.* 2d rev. English edition. New York: Macmillan. → First published in German. This edition was edited by Hilda Geiringer.

(1931) 1945 *Vorlesungen aus dem Gebiete der angewandten Mathematik.* Volume 1: Wahrscheinlichkeits-rechnung und ihre Anwendung in der Statistik und theoretischen Physik. New York: Rosenberg.

(1939) 1951 *Positivism: A Study in Human Understanding.* Cambridge, Mass.: Harvard Univ. Press. → First published as *Kleines Lehrbuch des Positivismus.*

1949 Three Remarks on the Theory of the Ideal Plastic Body. Pages 415-429 in *Reissner Anniversary Volume: Contributions to Applied Mechanics.* Ann Arbor, Mich.: Edwards.

*Mathematical Theory of Probability and Statistics.* Edited by Hilda Geiringer. New York: Academic Press, 1964. → Based upon lectures given in 1946.

*Selected Papers of Richard von Mises.* 2 vols. Providence, R.I.: American Mathematical Society, 1963-1964. → Contains von Mises’ writings first published between 1908 and 1954. Volume 1: *Geometry, Mechanics, Analysis.* Volume 2: *Probability and Statistics; General.* Includes a bibliography on pages 555-568 in Volume 2.

## SUPPLEMENTARY BIBLIOGRAPHY

Cramer, Harald 1953 Richard von Mises’ Work in Probability and Statistics. *Annals of Mathematical Statistics* 24:657-662. → Includes a selected bibliography of von Mises’ work.

Wald, Abraham (1938)1955 Die Widerspruchsfreiheit des Kollektivbegriffes. Pages 25-45 in Abraham Wald, *Selected Papers in Statistics and Probability.* New York: McGraw-Hill.

## Mises, Richard von

# MISES, RICHARD VON

(*b*. Lemberg, Austria [now Lvov, U.S.S.R.], 19 April 1883; *d*. Boston, Massachusetts, 14 July 1953)

*mathematics, mechanics, probability*.

Von Mises was the second son of Arthur Edler von Mises, a technical expert with the Austrian state railways, and Adele von Landau. His elder brother, Ludwig, became a prominent economist; the younger brother died in infancy. After earning his doctorate in Vienna in 1907, Richard taught at universities in Europe and Turkey and then, from 1939, in the United States. In 1944 he became Gordon McKay professor of aerodynamics and applied mathematics at Harvard. During his European period, he married Hilda Pollaczek-Geiringer, one of his pupils. Proud to call himself an applied mathematician, he was the founder and editor, from 1921 to 1933, of the wellknown *Zeitschrift für angewandte Mathematik und Mechanik*. He was a scholar with wide interests, who wrote perceptively on the philosophy of science from a positivist point of view, and who was also an authority on the poet Rilke.

Von Mises’ early preoccupation with fluid mechanics led him into aerodynamics and aeronautics, subjects that in the years immediately before 1914 had received a major fillip from the success of heavier-than-air flying machines. He himself learned to fly and in the summer of 1913 gave what is believed to be the first university course on the mechanics of powered flight. After the outbreak of World War I he helped develop an Austrian air arm, and in 1915 the team he led produced a giant 600-horsepower military plane with an original wing profile of his own design (wing theory was perhaps his specialty).

In 1916 he published a booklet on flight, under the auspices of the Luftfahrarsenal in Vienna. It went into many enlarged editions and is the basis of *Theory of Flight*, published with collaborators in English toward the end of World War II. Other, allied topics to which he contributed were elasticity, plasticity, and turbulence. He also worked in various branches of pure mathematics, particularly numerical analysis.

Von Mises’ concern with the border areas of mathematics and the experimental sciences was reflected in his giving much thought to probability and statistics. In 1919 he published two papers that, although little noticed at the time, inaugurated a new look at probability that was destined to become famous. The background to this contribution was the slow buildup, during the nineteenth century, of a frequency theory of probability, in contrast to the received classical theory of Laplace. The fathers of the frequency theory, Poisson in France and Ellis in England, had identified the probability of a given event in specified circumstances with the proportion of such events in a set of exactly similar circumstances, or trials. The weakness of this position is the necessary finiteness of the set, and there is no obvious way of extending the idea to those very large or infinite sets that in practice must be sampled for probabilistic information. The Cambridge logician John Venn improved the theory in 1866 by equating probability with the relative frequency of the event “in the long run,” thereby introducing a mathematical limit and the infinite set. Nevertheless, even this reformation failed to make the theory compelling enough to tempt mathematicians to put it into rigorous terms; and Keynes in his *Treatise on Probability* (1921) expressed his inability to assess the frequency theory adequately because it had never been unambiguously formulated. This was the deficiency that Von Mises attempted to correct.

What Von Mises did was to splice two familiar notions, that of the Venn limit and that of a random sequence of events. Let us consider the matter in terms of a binary trial, the outcome of which is either a “success” or otherwise. Given an endless sequence of such trials, in the sense of Bernoulli binomial sampling, what can we say about it probabilistically? A meaningful answer, said Von Mises, is possible only if we postulate (1) the mathematical existence of a limiting value to the fraction successes/trials, and (2) the invarianee of this limit for all possible infinite subsequences formed by any rule of place selection of trials that is independent of their outcomes. Then the limit can be called the probability of a success in the particular system. It then follows that the probability of a single event is formally meaningless; random sampling is a *sine qua nan*; and the sequence (otherwise collective or sample space) must be clearly defined before any discussion of probability—in this strictly operational sense—can be undertaken.

The intuitive appeal of Von Mises’ limiting frequency theory is strong, and its spirit has influenced all modern statisticians. Remarkably, however, the mathematics of the theory, even after sophistication by leading probabilists, has never been rendered widely acceptable, and some authorities today do not mention Von Mises. In advanced work, the measuretheoretic approach initiated by Kolmogorov in 1933 is most favored. On the practical side, his statistical writings suffered from a foible: he denied the importance of small-sample theory. Von Mises’ *Probability, Statistics, and Truth*, published in German in 1928 and in English in 1939, is not a pedagogic text but a semipopular account, very subjective in tone, good on the historic side, and in general notably stimulating.

## BIBLIOGRAPHY

**I.** Original Works. The core of Von Mises’ work is to be found in the following six books: *Probability, Statistics and Truth* (New York, 1939; 2nd ed., 1957); *Theory of Flight* (New York, 1945); *Positivism, a Study in Human Understanding* (Cambridge, Mass., 1951); *Mathematical Theory of Compressible Fluid Flow* (New York, 1958), completed by Hilda Geiringer and G. S. S. Ludford; *Selected Papers of Richard von Mises*, Philipp P. Frank *et al*., eds. (Providence, R.I., 1963); and *Mathematical Theory of Probability and Statistics* (New York, 1964), edited and complemented by Hilda Geiringer. His first papers on probability are “Fundamentalsätze der Wahrscheinlichkeitsrechnung,” in *Mathematische Zeitschrift*, **4** (1919), 1–97; and “Grundlagen der Wahrscheinlichkeitsrechnung,” *ibid*., **5** (1919), 52–99. A good bibliography of 143 works is in Garrett Birkhoff, Gustav Kuerti, and Gabor Szego, eds., *Studies in Mathematics and Mechanics Presented to Richard von Mises* (New York, 1954), which contains a portrait.

**II.** Secondary Literature. The opening chapters of *Mathematical Theory of Probability and Statistics* (see above) contain a survey by Hilda Geiringer of other workers’ developments of Von Mises’ controversial theory, as well as a synopsis of Kolmogorov’s rival theory and its relation to that of Von Mises. A critical essay review of this book by D. V. Lindley is in *Annals of Mathematical Statistics*, **37** (1966), 747–754. W. Kneale, *Probability and Induction* (London, 1949), marshals some logical arguments against limiting-frequency theories. On the other hand, H. Rcichenbach, *The Theory of Probability* (Berkeley, 1949); Rudolf Carnap, *Logical Foundations of Probability* (Chicago, 1950); and Karl Popper, *The Logic of Scientific Discorery* (New York, 1958), are all, in different ways and with various emphases, derivative and sympathetic.

Norman T. Gridgeman