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Social Science and Modernism: the Case of Karl Menger

Robert Leonard*

“[He had] an appreciation of cultural attainments and tolerance and respect for cultural differences, combined with a deep suspicion of rabid nationalism, particularly the German variety. Also a fascination with structure, whether artistic, scientific, philosophical, or theological, but a rejection of any aura of mysticism or mumbo-jumbo accompanying such structure” Editors’ introduction to Menger (1994), p.viii

Introduction Born in Vienna in 1902 and dying in Chicago in 1985, Karl Menger has been rightly described as a mathematician of considerable culture and erudition. Living in Vienna until 1937, he was witness to the energy of that city, which manifested itself not only in activity in a range of scientific, intellectual and artistic fields, but also great interaction between the often polyvalent specialists in those fields and the city’s professional and merchant elite. He is known to mathematicians for his original contributions to various areas of modern mathematics, such as the theory of curves and dimension, several fields of geometry, and his considerable efforts, particularly after his emigration to the U.S., in the development of mathematical pedagogy. He is remembered by historians of philosophy as an associate, albeit a somewhat reluctant one, of the Vienna Circle of Hahn, Schlick, Neurath and Carnap. By historians of economics, Menger is remembered as a catalyst in the Viennese chapter of the development of modern mathematical economics. Because of his family history, he had privileged access to economics circles in Vienna, and he became something of a conduit between the two “discourses” of economics and mathematics. In the 1930’s, this was an unusual role, for social science was not a subject of predilection amongst serious mathematicians, and most of the academic economists in Vienna were decidedly sceptical about treating their subject in a mathematical manner. Their reasons covered the gamut, ranging from the epistemological to the political, but tradition was also important, for none other than Carl Menger himself, biological father of the mathematician and spiritual father of the economics fraternity, had shown himself to be a mathematical sceptic. Menger Sr. had also insisted on the conceptual separation between the science of nature and the science of society, and argued, against the German historians, for the legitimacy of a universal, non-mathematical, economic theory. All of this had created the long-standing feeling in Vienna that economics was a unique and special sphere, contributing to the resistance towards formal treatment. Menger’s influence upon the economics fraternity was exerted in two ways. First of all, he participated in several of their seminars and discussion groups, such as those * Université du Québec à Montréal and Visiting Fellow, Center for the History of Political Economy, Duke University. This is a preliminary and incomplete draft, intended only for discussion at Duke. Please do not quote or cite without my permission.

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surrounding Ludwig von Mises and Hans Mayer, which met regularly in the city. Here, he acted as something of a fly in the ointment, pointing out cases of shoddy argument and challenging figures such as Mises for their lack of rigour or their inability to keep their politics out of their economic analysis. Secondly, as organiser of a somewhat similar group for mathematicians, his Mathematical Colloquium, he was able to promote an interest in economics among some of them. The most prominent example here was Abraham Wald and his contributions to the theory of Walrasian general equilibrium. Given the extent to which general equilibrium theory dominated the field in the second half of the 20th century, it is this aspect of Menger’s Viennese career that has received most attention amongst historians of economics.1 Menger also used his acquaintance with the economics fraternity in order to help his impoverished mathematics students. Thus Wald and Franz Alt would provide mathematics lessons to figures such as Karl Schlesinger, a private banker, and Oskar Morgenstern, of the Austrian Institute for Business Cycle Research. All of this helped to develop the relationship between these two previously disparate communities. Menger’s relationship with Morgenstern had another, less immediately obvious, influence on the field, namely in the stimulus it provided the economist as quasi-founder of game theory. Morgenstern’s status as “maverick” in Vienna of the 1930’s owed much to his Menger-inspired questioning of the logical coherence of economic theory, including the conception of the agent and his knowledge: if all agents were presumed to have perfect foresight, then what was to stop them trying to behave strategically and outguessing one another? With foresight so perfect, where would the outguessing stop and what did this mean for equilibrium? And how could one mathematically specify both this interaction and the resulting equilibrium? Morgenstern drew much sustenance from Menger in both developing this kind of critique and persisting in the belief that the solution remained mathematically tractable. Of particular importance here was Menger’s 1934 book on mathematical sociology, Moral, Wille und Weltgestaltung. Ostensibly a formal treatment of ethical decisions, this book was equally a mathematical analysis of social compatibility and group formation. While the “agents” involved in Menger’s theoretical schemes were simpler than those with which Morgenstern was struggling, the fact that they were shown, logically and rigorously, to be compatible at all, gave Morgenstern hope that the more complicated actors of economic theory could be similarly treated. It was thus “primed” that he fell, a few years later, into his collaboration with von Neumann at Princeton. As with the theory of general equilibrium so with game theory: given the extent to which the latter rose to prominence in the last decades of the 20th century, Menger becomes an important, if somewhat unwitting, contributor to the history of that field.2

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1 See the work of Roy Weintraub and Lionello Punzo. More recently, the Viennese mathematician, Karl Sigmund, has edited volumes of both the Colloquium’s proceedings and Menger’s own writings (Menger 1998). 2 See Leonard (1998) and forthcoming.

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From the beginning of my reading and archival work on Menger, both at the Illinois Institute of Technology and here at Duke, I noticed his frequent references to the sphere of art and aesthetics. While this is something that is discussed only occasionally in the correspondence I have examined thus far, much of which is taken up with detailed mathematical matters, it does feature in his posthumously published Reminiscences (1994), in both their archival draft- and published versions. At some points in his publications too, aesthetic considerations are important. Finally, the visual arts occupied a significant place in Menger’s life. All of this, taken together, seemed to point towards something of greater than incidental importance, worthy of closer examination.3 Menger was actively interested in modern art of various kinds and, if one is to judge by various obiter dicta, he was also self-consciously Modern – aware of the novelty and perhaps even slightly scandalous nature of his artistic tastes. In the mid-1920’s, when he was in his early 20’s, he developed an interest for the work of certain graphic artists of the period, including those by the Duch avant-garde. In the period 1925-27, not only did Menger live in The Netherlands, he spent the first year or so in Laren, an artists’ colony outside Amsterdam, where his then-mentor Ludwig Brouwer lived. During that time, he travelled to Paris to see Piet Mondrian’s studio, and at some point he acquired for his own Vienna apartment modern furniture in the style of that made by De Stijl. In Vienna, he discussed art with Clara Wittgenstein, the philosopher’s aunt, and he visited, and admired, the Stonborough House, which Ludwig had designed for his sister Margarethe. He spent the academic year 1930-31 in the U.S., first at Harvard, then at the Rice Institute in Texas. In Cambridge, Massachussetts, he engaged both Harvard’s George Birkhoff and MIT’s Norbert Wiener in discussions of aesthetics, a subject on which the former was actively working at the time. Travelling across the U.S., Menger was greatly taken by the Native art of the American Southwest. Not only was Menger interested in art, but he drew connections between mathematical and artistic creativity. It would not be an exaggeration to say that he saw mathematics as an art form, albeit one that had its rules and often showed itself to be relevant to science. In Vienna in the late 1920’s and early 1930’s, when defending complete freedom in the pursuit of abstract mathematics in the context of the foundational debates, he invoked the

3 It is interesting to observe how ideas evolve. In one or two unpublished papers in the mid-1990’s, I began groping towards an understanding of the place of art in the economics and social science of interwar Vienna. However, in the published work that eventually grew out of that (with the exception of a paper on Neurath), the artistic dimension essentially disappeared, focused, as it was, on developments in economics and game theory. In the latter stages of my work on that, however, particularly through my explorations of von Neumann, I found myself once again confronted with questions of aesthetics and style. This time, it was style in mathematics and the manner in which it assumed political importance in Germany in the 1930’s. In short, I learned to see von Neumann’s game theory not merely as something stimulated by the politics of the late 1930’s but as the reassertion of a “Modern”, Hilbertian style in the face of such political chaos, an idea I now see to be quite congruent with Jeremy Gray’s new broad portrayal of mathematical Modernism. All of this has given me fresh impetus to return to exploring the social sciences in relation to the various expressions of Modernism in the first half of the 20th century.

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very Modern defence of “Art for Art’s Sake”. When he opposed Otto Neurath’s campaign for Unified Science, it was not because of any belief in a distinction between the natural and social sciences (regardless of what Menger Sr. may have written), but because he believed that trying to view mathematics as a sharply-defined “science” was to ignore the artistic nature of mathematical and scientific creativity. When Menger moved to the U.S., he soon left the University of Notre Dame for the Illinois Institute of Technology, whose campus was then being built by Mies van der Rohe, and the architecture of which Menger likened to the work of Mondrian. In 1943, with encouragement from Walter Gropius, Menger’s wife Hilda developed a children’s game in architectural blocks, designed as an aid in cultural and political education. In 1952, at the Chicago’s Museum of Science and Industry, Menger organised a geometry exhibition, another “aesthetic space”, not unreminiscent of Mondrian’s studio, designed in this case to present the wonders of geometry to a broader audience. This essay is a tentative and preliminary attempt to explore Menger from an aesthetical point of view, to examine him from the perspective of the psychology of style. In a series of interconnected vignettes, we consider his engagement with the realm of artistic creation, looking at the places he lived, the things with which he surrounded himself, with a view to understanding how his imagination worked. We see the resonance between the geometrical structures he explored as a mathematician, some of which could be seen by the eye, and the structures he admired in Modern art. We see in Menger a concern for simplicity, clarity and sharp definition, aesthetic qualities that, for him, had political ramifications. We consider the abstract theoretical structures he construed in his 1934 book in sociology, and probe the psychological, aesthetic and political dimensions of that work. Sickness and the sanatorium In a recent article in Medical History, Margaret Campbell writes about “What Tuberculosis did for Modernism”, examining the relationship between the curative environment and modernist design.4 The sanatorium movement emerged in the late 19th century and strengthened with the 1882 discovery of Mycobacterium tuberculosis. At institutions such as Hermann Brehmer’s Görbersdorf Sanatorium, set in sylvan tranquility, patients followed a cure centred upon the “jour médical”: “good food, rest and gentle exercise . . . [and] the obligatory daily two-hour period of rest in the open air, usually taken between two and four o’clock in the afternoon, and frequently observed in silence” (Campbell, p 465). Because of the need to recline in the fresh air, says Campbell, “certain architectural requirements were essential to early sanatorium design” (ibid), such as deep verandahs, balconies and covered corridors, all furnished with reclining couches. Many of the noted Modern architects and designers, such as Walter Gropius, Marcel Breuer, Le Corbusier and Mies van der Rohe, were associated with the development of hygienic housing and sanatoria. For example, the Weissenhof Seidlung housing exhibition in Stuttgart in 1927 promoted greater light and hygiene for urban dwellers. Because it was believed that

4 Campbell (2005).

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sunlight destroyed the TB bacillus, the need to accomodate reclining patients promoted the use in sanatoria of flat roofs and large terraces and balconies. At Hilversum in The Netherlands, the name of the Zonnestraal (sunbeam) sanatorium, built by De Stijl architect Jan Duiker, reflected not only the therapeutic requirement but also the place of sunlight in the theosophical writings of M.J.H. Schoenmaekers. This Catholic priest and mathematician-turned-mystic was an important influence on some members of De Stijl, with his ideas on “plastic mathematics”, namely the spiritual roles of the horizontal line and the vertical ray, and their juxtaposition in the form of the cross. Thus, Duiker, describing his own architecture, could speak of its “spiritual economy. . . leading to dematerialisation, spiritualisation. It is undoubtedly the intrinsic economy of the material that enables us to achieve more and to satisfy higher spiritual demands more truly than our forefathers were able to” (quoted by Campbell, p.470). Other new sanatoria incorporating modern design were built, says Campbell, in Davos, Switzerland (Klinik Clavadel, 1931-1933) and in Paimio, Finland (1929-1933). Elsewhere, many existing sanatoria were modified in order to acommodate increased numbers of patients as they lay outdoors, with balconies or flat-roofed, terraced extensions being added. The “cure” also saw the appearance of reclining chairs, in Modernist style, built of bentwood, laminated wood or chrome. Many were designed by Aalto, Breuer and Le Corbusier, and were soon “considered appropriate relaxing chairs for the modernist house” (p. 481). When they entered mass-production, these chairs were made especially by the Viennese firms of Gebrüder Thonet and Jacob & Josef Kohn. The famous couch at the Davos sanatorium was immortalized, like the sanatorium itself, by Thomas Mann in The Magic Mountain. Karl Menger spent several of his formative years at such a sanatorium, the Hoflacker Kurhaus in Aflenz, in Styria, in the Austrian Alps. Although the building was not in the strict Modernist style, a postcard from the mid-1920’s shows that it had been extended in the manner described by Campbell, with a flat-roofed wing with terraces off to the left, and shaded balconies elsewhere.

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Having being bed-ridden at home for several months, Menger entered the sanatorium in the autumn of 1921 and stayed till the spring of 1923. When he entered, Menger had already made his mark as a capable student of mathematics, doing original work in the theory of curves in a 1921 seminar by Hans Hahn. Although Menger would later remember his confrontation with TB as a “year” well-spent, reading philosophy and mathematics, it seems to have been more difficult than that. For during his time at the sanatorium, Menger lost his father. According to Karl Sigmund, Menger also had literary ambitions, working for three years, from 1920, on an abortive play about La Papeesa, the apocryphal female Pope Joan of the 9th century, and maintaining a correspondence with Arthur Schnitzler, who apparently thought little of his literary skills.5 Although Menger was cured of TB, the “kurhaus” would remain a feature of his life. Especially in moments of crisis and difficulty, he would return periodically to Aflenz, or to the other magic mountain, Semmering, more famous haunt of artistic Vienna.6 In the summer of 1925, when his mother passed away, he returned from Holland to Vienna, had a complete nervous breakdown and, once again, had to spent a significant period at the sanatorium.7 From Vienna to Laren It was not illness that brought Menger in contact with De Stijl, but rather his contact with Otto Neurath:

5 See Sigmund (?). 6 Thomas Mann and Kafka are the defining figures in Menger’s recollections of Aflenz: while he was “recuperating on one of Austria’s magic mountains, Franz Kafka was dying on the other one” (1994, p. 46). In fact, Kafka died, not on the Semmering, but at Kierling, a low-altitude sanatorium close to Vienna, in 1924. See Gilman (1995), p. 239. 7 See Van Dalen, Vol. II, p. 512.

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“The Neuraths lived in an old building located in a workers’ district of Vienna, where Otto kept his extensive library. In the years 1923/25, I went to their picturesque apartment a few times with my friends Otto Schreier and Witold Hurewicz who were to become distinguished mathematicians. Felix Kaufmann had introduced us. In these totally informal meetings, we had stimulating conversations with Otto; but all in all, we mathematicans felt more drawn to the quiet Olga. While strongly individualistic himself, he was, as a matter of principle, even more strongly opposed to individualism. He always regarded himself as expressing ideas and feelings of the masses. His interest in statistics fitted in that general frame and so did of course his literary taste. Once he recommended to me to read Jack London with the words: “Great stuff. On proletarian concern”. (Reminiscences, Unpublished draft, Box 22, Vienna Circle Folders).

In the published version, Menger continues:

“In the late 1920’s, Neurath became greatly interested in contemporary graphic art; and in this field our interests met. He wanted to utilize that kind of art for what he called Bildstatistik (Pictorial Statistics). . . He invited excellent graphical artists, among them the German Gerd Arntz and the Dutchman Peter Alma, - the Belgian Frans Masereel he considered too romantic – and commissioned them to design the units . . . And he promoted this Bildstatistik with his full enthusiasm and energy first in Vienna, later in Amsterdam, and finally in London, where he founded the Isotype Institute” (Reminiscences, p. 62).

The artist considered too romantic by Neurath was Frans Masereel (1889-1972), Belgian author of highly imaginative graphic novels in exquisite woodcuts, including Le Soleil, Les Heures and Die Stadt. Masereel was a chronicler of everyday urban life, with an element of social critique running throughout much of his work.8 His books were very popular and they were admired by other writers, including Hermann Hesse, Thomas Mann and Romain Rolland. Menger clearly loved Masereel’s work, Le Soleil being, amongst other things, his gift of choice for Clara Wittgenstein, the philosopher’s aunt, whom he knew through his family.9

8 Castelman (1983) suggests that the ‘intensely political motivation” of Masereel’s work served to detract from its potential as an artistic endeavour (p.7). 9 Menger remembered the effect that the gift of Masereel had had: “Only once in our many conversations over the years did I discern anything that seemed to me to suggest her age. When I gave her Frans Masereel’s Le Soleil, a novel in woodcuts, which aroused the enthusiasm of young people and even the admiration of Thomas Mann, Hermann Hesse, and some of their generation, she seemed to praise his ingenious work only out of politeness” (Reminiscences, p.78)

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It is not insignificant that Menger could evince mild condenscendence at the “strong individualism” of Neurath, the putative socialist, and yet admire the radical artwork of Masereel. For while he had a genuine aesthetic appreciation for the graphic art of the period, Menger did not share the underlying radicalism. The black-and-white clarity and the extraordinary skill appear to have appealed to him, but not the call to revolution. Apart from a quiet admiration for the achievements of the educational programmes of Viennese Social Democrats, Menger appears to have been quite liberal in his politics – an attitude that extended to his views on mathematics. Menger would work aesthetic considerations into his mathematical and political engagements in a decidedly non-caricatural manner. The Bildstatistik being promoted by Neurath were those for use in what would become, in 1924, the Economic and Social Museum of Vienna, where measurable economic and social information was presented by pictorial means.10 While Menger was more inclined towards his father’s liberal theory of economic order than the centralised planning being promoted by Neurath, he understood the visual representations well and he liked them:

“In most traditional pictorial comparisons, each quantity had been represented by a picture of single entity; for example, each nation by the picture of a single man – small nations by smaller men. A quantity three quarters of another was represented by a picture supposedly three quarters the size of the other. But, Neurath asked, exactly what is “the size” of a picture? Its height? Its area? The volume of the pictured object? In dealing with two nations whose populations are 80 million and 60 million Neurath proposed an unequivocal representation by pictures of two groups of men in the proper numerical ratio: 8 and 6 men or 4 and 3 or 16 and 12 or even 10 and 7½. Neurath later referred to the representation of quantities by groups of equal typical figures as the method of isotypes (a name suggested, I believe, by Marie Reidemeister, later Mrs. Neurath). The idea itself was not really very original; representation by isotype had been devised before Neurath, though frequently alternating with inferior pictorial methods. But Neurath used the method exclusively, developed it systematically, and carried it out with great taste” (Op cit, p. 7.8).

The precise, yet expressive, Isotype figures were prepared by Gerd Artnz, formerly associated with the radical art group, Stupid, in Cologne. He became Neurath’s right-hand man, and would eventually move with him from Vienna to The Hague, when the Dollfuss regime closed down Neurath’s operations in visual education. The artist to whom Menger appears have been closest at the time was Peter Alma. Working both with paint and in woodcuts, Alma was influenced by Léger, and De Stijl artists Piet Mondrian and Bart van der Leck. A member of the Dutch Communist party, he had already visited the U.S.S.R in 1921, making contact with the Russian Constructivists and Suprematists, including Kandinsky, Lissitsky, Tatlin and Malevich.

10 For a pictorial exploration of Neurath, see Vossoughian (2008).

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Upon returning, he made contact with Gerd Arntz and would ultimately complete several tours of duty at Isostat, the Moscow extension of Neurath’s Vienna operation. In his woodcuts and paintings, Alma produced deliberately flat images, often overflowing the frame, and with most if not all perspective eliminated. His work was more consistently political than Masereel’s, portraying faceless figures of power, the dejection of the unemployed and the physicality of labour. If Menger liked this work, it was not for its political didacticism, but for the way in which it resonated with his experience as a mathematician: “Along with the great aesthetical enjoyment that I derived from the work of the graphical artists of the 1920’s, I also had the feeling that their work was the artistic counterpart of the tendency to abstraction in mathematics just as impressionism and the philosophy of Mach have been called artistic-scientific counterparts. Abstraction began to pervade mathematics in the first decade of this century” (Reminiscences, p. 62) Menger goes on to describe the development of algebra in abstract fields (as opposed to concrete fields of numbers); the generalization of the idea of space by Fréchet, who selected a few essential properties of Euclidean, non-Euclidean and other spaces in order to create abstract metric spaces, which Menger himself used in his theories of curve and dimension; and his own development of metric geometry, with theories of betweeness, geodesic line, and curvature of curves and surfaces. “Analogously”, he continues, “the graphic artists of the 1920’s selected only a few absolutely essential features of types of objects and people and represented, as it were, the Platonic ideas of those objects by these abstractions”(Ibid). In his work in various kinds of abstract geometries, concerned with the properties of abstract structures involving lines and planes, curves and point-sets, Menger fell squarely in the development of what Jeremy Gray has described as the “modernist transformation” of mathematics over the period 1890-1930. This evolution marked the emergence in that discipline, as in painting and literature, of “an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated – indeed, anxious – rather than a naïve relationship with the day-to-day world” (2008, p.1). At the beginning of the century, says Gray, in line with the increased professionalization and autonomy of the discipline, mathematics underwent a decisive shift. This found expression in both ontology, with the emergence of new autonomous fields of mathematics, and epistemology, with an emphasis on formal proof and a distrust of naive intuition, especially in geometry. “The changes in mathematics

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around 1900”, says Gray, “constitute a modernism, much as contemporary changes in art, literature or music do” (p.4). The exemplary work here was Hilbert’s Grundlagen der Geometrie: “The break with any kind of geometric intuition in defining geometric terms in characteristically modernist. . . A similar shift took place in the presentation of projective geometry” (p. 19). Themes that were central to modernism in art manifested themselves in the discipline of mathematics, including anxiety, about the relationship of the created objects to “the world”, and a deep interest in the history of the subject, often used in order to legitimize the new style. In the work he did and the manner in which he came to defend his practice, Menger can be connected to virtually every aspect of Gray’s characterization of mathematical Modernism. He is unusual among mathematicians, and interesting for us, insofar as he actually reached out to Modern art and sought to apply mathematics to the social realm.

* * * Had he not died in 1985, Menger would no doubt have appreciated the unwitting irony shown by Carel Blotkamp in her book of the following year on the Dutch modern art movement, De Stijl. To the writings of van Doesburg, Mondrian, Vantongerloo and other members of the movement, she says, one can apply what has been said of the school surrounding Dutch mathematician Brouwer: “The primary revelations of the creed are veiled in Dutch language. German and English expositions are available but it is said by converts with only expert knowledge in both the languages and the mathematics that only those who can think in Dutch can grasp the finer shades of meaning”.11 “[E]ven we Dutch authors”, Blotkamp continues, “have not the slightest idea, after half a century, of the meanings of certain statements made by these elated men” (p.xi). In March 1925, Menger left Vienna to go to The Netherlands on a postdoctoral stay that ended up lasting almost two years. There, he encountered the Dutch mysticism referred to by Blotkamp, in both their Brouwerian and De Stijl manifestations. With neither of them could he agree, as a true positivist, but only one would he perceive as truly threatening. By 1925, the village of Laren in the ‘t Gooi region outside Amsterdam was well established as an artists’ colony. Because of the attractive countryside and proximity to the capital, the area had attracted painters since the 1890’s, including William Singer, scion of the Pittsburgh steel family, who had renounced his inheritance and settled there in order to paint.12 Although all proved to be relatively short-lived, there were a number of Thoreau-inspired communes in the area, including one, Walden, established by Dutch psychiatrist Frederick van Eeden. Part of this general pastoral movement, mathematician L.E.J. Brouwer and his wife had built a rural cottage in Blaricum, adjacent to Laren, and had lived there since 1904. The couple lived according to a rather Spartan regime, which involved vegetarianism, outdoor exercise, periods of fasting, sunbathing, and, when possible, sleeping in the open air. By the time Menger arrived as postdoctoral student, Brouwer was a well-travelled mathematician with an international reputation, but he remained resolutely attached to his lifestyle in this setting of rural tranquility.

11 Blotkamp’s quotation is taken from E.T. Bell’s well-known, and contested, history, Men of mathematics. 12 See van Dalen (1999), pp. 59-64.

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Laren was also a significant geographical location in the development of the De Stijl movement in Modern Dutch art. Attracted by the presence of artists and other creative individuals, such as the composer Domsaeler and the previously mentioned M.H.J. Schoenmaekers, various members of De Stijl including Piet Mondrian and architect/designer Robert van’t Hoff lived at Laren at various times. In 1925, when Menger arrived, Mondrian had been back in Paris for six years, but both Bart van der Leck and Peter Alma were still living there. The mystical vein in Brouwer’s work came initially through the influence of Dutch Hegelian, G.J.P. Bolland, who held a chair at Leiden and was the dominant figure in Dutch philosophy at the beginning of the century. Brouwer was initially unsure whether he would become a philosopher or a mathematician, and he wrote several philosophical pieces, including his infamous 1905 Delft lectures, Leven, Kunst en Mystiek (Life, Art and Mysticism). As Brouwer’s biographer, Van Dalen, indicates, the central message of this essay is “the mystical doctrine that man’s ultimate goal and challenge is total introspection – a turning into oneself” (p.66). Everything else flowed out of that central idea. In Holland, said Brouwer, there had been an original earthly balance or equilibrium in human society. Then power struggles emerged between its members, leading to the corruption of the individual’s spirit, the pursuit of illusory goals through hard labour, and the destruction of the environment. Drawing on several mystical writers, including Meister Eckhart, Brouwer explores how introspection can lead the Self to victory over the sad World. Besides passages on the destruction of the natural environment, there are others portraying woman as the temptress, endangering male purity, all of which were for decades a source of embarrassment in Brouwerian circles.13 A final section on “Economics” insists that, without “foolishness and injustice”, the field would not exist. The desire for things that makes economics relevant is born of vanity and the thirst for power, both of which run contrary to a life of introspection. By the time Menger read this, he was already falling out with Brouwer, and this for two main reasons. First of all, he fell into an ugly argument with him over priority in the field of dimension theory, accusing Brouwer of actually tampering with the historical record.14 Once this row got under way, it lasted for several years, continuing well after Menger’s return to Vienna in 1927. Secondly, and inseparably, Menger developed a disliking for Brouwer’s philosophical ideas on the foundations of mathematics, the debates over which were then raging in the discipline. Invoking arguments about a primordial “Ur-intuition” to which the natural numbers and counting were central, Brouwer had taken a position against the application of the law of the excluded middle to infinite sets, the argument being that since it is not humanly possible to check all the elements of such a set, one can not simply assume that the law of the excluded middle applies to all its members. The reason that Brouwer’s position was so controversial was not because of its seeming idiosyncracy, but because, were it to be taken seriously, it would imply the dismissal of very large parts of mathematics dependent on non-constructive proofs which invoked the

13 Not until 1996 was this essay translated into English: see Brouwer (1905). 14 Van Dalen devotes a substantial section in Vol. II of the Brouwer biography to this priority debate. The Menger archives contain several documents relevant to the debate, not examined by Van Dalen, including Menger’s copies of the documents with which he believed Brouwer was guilty of editorial interference.

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law of the excluded middle. Menger would soon turn on Brouwer, and within a short time of returning to Vienna, he would accuse him of mysticism and draw on aesthetic arguments in order to rebuff him and defend modern mathematics. The other area where Menger encountered mysticism was more benign, for here, unlike in mathematics, it resulted in the elaboration of pleasing structures, not their destruction. If Masereel, Artnz and Alma stopped well short of achieving full abstraction in their artwork, the De Stijl movement is remembered for its achievement of a purely abstract, often geometric style. This is not the place to enter upon a discussion of the finer distinctions between Van Doesburg, van der Leck, Huszár, Vantongerloo or Mondrian. Suffice it to mention that Bart van der Leck began to create abstract forms by painting out large sections of figurative subjects, leaving rectangular or polygonal fragments on the canvas (from which one could sometimes guess the underlying object). Mondrian proceded towards abstraction by, initially, “fragmenting” the façades of the churches and buildings he was painting (see Composition in Oval with Colour Planes, 1, 1914). Unlike the cubists, whom he would ultimately regard as not going far enough in their degree of abstraction, Mondrian stuck to representing one façade only, and that of mainly outdoor subjects. As Els Hoek (1986) shows, there were already mystical underpinnings to Mondrian’s work when, in 1908, aged 36, he began to deform shapes and use non-realistic colours. He became familiar with the writings of Madame Blavatsky and Viennese mystic Rudolf Steiner, and joined the Theosophical Society. Thus the presence of the oval in his early abstractions was related to his interest in the androgyne myth, found in Theosophy and Hindu mythology, according to which man stemmed from a “cosmic egg” in which both sexes were united.15 Mondrian also drew on the Laren philosopher Schoenmaekers’ mystical writings on the balance between the horizontal (female) and the vertical (male) line. The term Neo-Plastic, which he used to describe his painting, was inspired by Schoenmaekers. The aforementioned Bolland was also important to Mondrian, and he quotes him in his early writings in the De Stijl review: “The beautiful is the true in the perceptual mode. And truth is a multiple unity of opposites . . . The concept of beauty is a relational one – of aesthetic relations, of perceptually agreeable and thus emotionally satisfying

15 Hoek notes that this interest in the male-female distinction motivated the work of Chagall and Brancusi, and was simulated by the surge of interest in Oriental religions at the beginning of the century (op cit, p.44ff).

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relationships” (Hoek, p. 48, originally in De Stijl, Vol. 1, No. 9, p.103). Throughout the late 1910’s, along with other members of De Stijl, Mondrian moved towards increased abstraction, developing a “geometric form language” (p.50). The oval form faded away and he went from presenting fragmented facades to juxtaposing rectangles of colour, and from there to introducing a grid in which the colours were then placed. He struggled with the structure of the grid, finally settling upon a network of non-deviating lines as a host frame.16 Underlying Mondrian’s work throughout was the idea that a piece of art sought to give expression to the higher reality that lay behind the appearance of things. In The Secret Doctrine, by Madame Blavatsky, he found confirmation of his belief that “rectangular enclosure was the most exact expression of equilibrium” (p. 59). (Composition with Red, Yellow, Blue and Black, 1921). Upon returning to Paris in 1919, Mondrian occupied temporary quarters until finally moving into his own studio at 26 rue du Départ, behind the Gare Montparnasse.17 There, he set about the shaping his complete environment along neo-Plastic lines, painting the walls in a deliberate way with large panels in carefully chosen colours, and strategically placing on them smaller rectangles of colour. Window openings and doors were carefully integrated into the whole, in order to give an impression of balance and harmony. The studio attracted many visitors, including Ben Nicholson and Alexander Calder. Apparently the latter’s invention of sculptural mobiles was stimulated by his visit to Mondrian’s studio.

Karl Menger also visited, having been guided to the studio by Alma back in Laren. In the autumn of 1926, to his girlfriend “Mitzi” in Vienna, Menger reported: “I liked Paris so much that I would like to return if it is at all possible. Just in passing, and to annoy you, I also visited Mondrian the painter, who paints only squares. He showed me photos of all his pictures; the development is quite interesting. Also his studio, whose walls are

covered with large rectangles in different colours and sizes. I liked it very much” (Karl

16 This approach was criticised by van Doesburg and Huszár. 17 See Bois (1982).

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Menger to Hilda Axamit, undated, Karl Menger Papers, Duke University, Box 1, Folder: Undated). The depth of Menger’s engagement with the De Stijl movement remains to be clarified. It is not clear to what extent he read their magazine, which ran from 1918 till Van Doesburg’s death 1931, and provides, amongst other things, an extended discussion of the relationship between this new abstract art and science and mathematics, including Mondrian’s sometimes unintelligible justifications of Neo-Plasticism. Nonetheless, it is clear that Menger was influenced by De Stijl, its mystical underpinnings notwithstanding. Not only would he refer fondly to Mondrian many decades later, but at some stage in the interim, he decorated his Vienna apartment in the modern style, designing his own furniture, and having it specially made. The surviving photographs include a spare geometric cabinet exactly like that by De Stijl designer Robert van’t Hoff, featured in the magazine in 1927, and an easy chair in tubular steel and rattan. On the other hand, the mysticism of the now demonized Brouwer would become a weapon to be used against him. Before considering that, however, let us follow Menger to America. “On the Trip of an Austrian Professor into Eldorado” It was armed with an essay thus named that Menger set off for the U.S. in the fall of 1930. Clara Wittgenstein had given him a copy of this witty 1905 travelogue, written by Ludwig Boltzmann after his sojourn in California. Menger visited Harvard from Sept. 1930 till February 1931, lecturing on the theory of curves and dimension. At the time, George Birkhoff was chairman of the mathematics department, and perhaps the most influential figure in North American mathematics. In his recollections, Menger singled out the Birkhoff’s interest in art:

“Birkhoff was intensely interested in aesthetics at that time. He had just returned from a world tour and brought home interesting phonograph records of various types of Oriental music. While the experience had certainly not made him an aesthetic relativist, there were some positivistic elements in his aesthetic studies. Steering clear of what he called “philosophical definitions” of art (such as Croce’s “the expression of an impression”) Birkhoff tested his theory of beauty. He let groups of students rank polygonal figures according to beauty and compared the results with the ranking according to his formula. The latter was based on the idea that, roughly speaking, beauty increases with order and decreases with complexity” (Reminiscences, p.158).

Birkhoff’s foray into aesthetics at this time culminated in the publication of his 1933 book, Aesthetic Measure. Here, he explains his early interest in the “quasi-mathematical” riddle of melody and musical form, and his return to the subject in 1924, when he found that his nascent theory was best applied to visual objects and geometric forms, such as

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“polygons, tilings, and vases” (p. vii). Then, in 1928, he spent a half-year in the Far East and Europe compiling further material for examination.18 The formula contrived by Birkhoff to measure aesthetic value was M = O/C, which related M, an object’s aesthetic measure; O its order, and C its complexity. Complexity was determined by the effort required by the brain in apprehending the object in question. This effort took the form of feelings of tension as the brain made the required sequence of automatic adjustments demanded by the apprehension. The sum of these (measurable) indices of brain tension yield C, a measure of complexity. Thus, Birkhoff says, a convex polygonal tile is very quickly seized by the viewer. Nonetheless, the viewer has to scan each side, bringing “a slight feeling of tension attendant upon each adjustment, and the complexity C will be measured by the number of sides of the polygon” (p. 6).19 Registering complexity is different from registering aesthetic feeling. For the latter, Birkhoff draws on the work of German psychologist, Theodor Lipps, whose Ästhetik: Psychologies des Schönen und der Kunst (2 vols. 1903 and 1906), emphasized the viewer’s ability to identify, or empathize with, the work of art being viewed. “In many cases of aesthetic perception, say Birkhoff, there is more or less complete identification of the percipient with the aesthetic object”(p.6).20 For example, symmetrical objects possess “exceptional and desirable qualities” and thus tended to evoke “an intuitive aesthetic feeling of positive tone” (p. 7). These intuitive associations Birkhoff divided into two kinds, formal and connotative. Formal assocations“involve reference to some simple physical property of the object” (p.9) and are immediately obvious. Symmetry in a geometrical figure is a prominent example, says Birkhoff. Connotative associations are all that are not formal. More difficult to identify, they depend on personal experience. Birkhoff terms “elements of order” the properties of an object that produce these formal and connotative associations. Depending upon the aesthetic object in question, the positive formal elements of order may include “repetition, similarity, contrast, equality, symmetry, balance, and sequence” (p.9). Centres of interest and repose are also important. For example, “a painting should have one predominant center of interest on which the eye can rest” (p.10). Negative elements, he says, include ambiguity, undue repetition, and unnecessary imperfection, an example of the latter being a rectangle that is nearly but not quite a square. As with complexity, Birkhoff makes order measurable by aggregating the sum of observed indices of order (positive, zero or negative). Thus, in the case of a polygonal tile, symmetry about the vertical would count as a positive index, as would symmetry of rotation, or the coincidence of the sides of the polygon with a rectangular network, as in the case of the Greek cross (figure 9 in Plate II above).

18 See also Birkhoff (1929), (1931a,b,c). The accuracy of Menger’s recollection about the polygons suggests that he may well have acquired a copy of Birkhoff’s book. 19 For a simple musical melody, Birkhoff suggests that the index of complexity will be the number of notes it contains. 20 It is worth noting here that Lipps’ work was an important point of reference for another German work to which Birkhoff makes no reference, namely Wilhelm Worringer’s (1908) essay, Abstraction and Empathy. See below.

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Significantly, Birkhoff, on a later page, notes that his analysis cannot take account of the connotative elements, they being “of inconceivable variety and . . . beyond the range of precise analysis” (p.13). Having discussed several possibilities for relating C and O, Birkhoff settles on the simple ratio of the two as the aesthetic measure. He argues this using the analogy to business profitability. Amongst a group of firms of a single type, the most successful one is that with the highest ratio of profit to investment. He notes that it is impossible to use the measure to compare different types of aesthetic object (e.g., an image and a melody). Also, given the variety of preferences amongst individuals, according to their taste and aesthetic experience, it is intended as a measure valid for the “normal or average judgment of some selected group of observers” (p.11). Birkhoff devotes chapters to the examination of polygonal forms, ornaments and tilings, vases, music, and poetry. Polygons, which he had had his students view and evaluate, are simple, he says, and are relatively free of connotative elements or order. He explores the importance of symmetry of various kinds (about a line or a point) in different triangles and symmetric quadrilaterals. He also refers to Fechner’s (1876) experiments on rectangles, which showed the Golden Rectangle to be favoured. Figures that could be inscribed in a rectangular network, such as the Greek cross, were deemed aesthetically significant. So too, but less so, were those that fitted into a diamond-shaped network. In the case of the crosses and the Swastika, connotative factors, beyond the scope of the theory, were likely to be significant. In applying the formula to polygons, order, O, was calculated as the sum of a range of values contributed by particular features: V: vertical symmetry (+) E: equilibrium (+) R: rotational symmetry (+) HV: relation to horizontal-vertical network (+) F: unsatisfactory form (-) Equilbrium, for example, referred to optical equilibrium. Complete optical equilibrium required that certain rather complicated and, by Birkhoff’s admission, arbitrary conditions be met, namely that the centre of area of the polygon lie between two vertical lines passing between the extreme points and at a distance greater than one sixth of the total horizontal width of the polygon. The conditions determining satisfactoriness of form, and therefore F=0, were also complicated, including the stipulation that “the minimum distance from any vertex to any other vertex or side or between parallel sides is at least 1/10 the maximum distance between points of the polygon” (p.44). In the summer of 1929, at Columbia, and again in the summer of 1930, at Harvard, just before Menger’s arrival, Birkhoff had conducted experiments with students, who judged the aesthetic value of the polygons. The experimental results “were found to be in substantial agreement with the arrangement obtained by the formula” (pp. 45-6). Of 90 polygons examined by Birkhoff, the top 15 in terms of aesthetic value were the following:

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While Menger would have agreed with Birkhoff’s dismissal of all mystical explanations of aesthetic pleasure, his reaction to the mathematician’s work was critical. He was clearly scornful of the fact that “highest in Birkhoff’s order of beauty of polygons stood a square and a rectangle with sides parallel to the edges of the paper (the longer sides of the rectangle being horizontal for the sake of “stability”), and an equilateral triangle with a horizontal base” (Reminiscences, p. 159). Birkhoff, he felt, had “declared the ratio of order to complexity as the measure of beauty without, in my opinon, sufficiently stressing the arbitrary character of this stipulation. Of course, he was aware of the need for numerical evaluations of order and complexity as a prerequisite for a discussion of their quotient; but in assigning such values he could not help introducing numerous further

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elements of arbitrariness. Certainly his result was far from reflecting my own aesthetical feelings” (op cit, pp. 158-9). Nor would he have likely agreed with Birkhoff’s attempt at qualitative application of the elements of his quantitative theory. In a closing chapter on “Art and Aesthetics”, Birkhoff acknowledges that the limitation of his study to the auditory and visual domains is very restrictive as far as aesthetics is concerned: “for instance, a system of laws may be beautiful, or a mathematical proof may be elegant, although no auditory or visual experience is directly involved in either case. It would seem indeed that all feeling of desirability which is more than mere appetite has some claim to be regarded as aesthetic feeling” (p. 209). Birkhoff’s measure was based upon the physiological interpretation of aesthetic appreciation: “an aesthetic field corresponds to a network of associative nerve fibres in the brain” (p.210). It might be possible, he thought, to use similar factors in a qualitative way, and apply them to art forms where the network of assocations was less established than they were in the evaluation of polygons. Thus, well known aesthetic maxims might be interpretable in terms of the concepts of complexity and order, viz., “(1) Unify as far as possible without loss of variety (that is, diminish the complexity C without decrease of the order O) (2) Achieve variety in so far as possible without loss of unity (that is, increase O without increase of C). (3) This ‘unity of variety’ must be found in the several parts as well as in the whole (that is, the order and complexity of the parts enter into the order and complexity of the whole). From our standpoint such maxims are all contained implicitly in the aesthetic formula M = O/C” (p. 211). In this connection, Birkhoff refers to the work of two Harvard colleagues, Denman W. Ross and Arthur Pope. According to Ross’s (1907) Theory of Pure Design, it was the combination of Harmony, Balance and Rhythm that produced “Order”, the concept that became fundamental in Birkhoff’s analysis.21 Birkhoff borrows from Pope’s (1929) An Introduction to the Language of Drawing and Painting; I, The Painter’s Terms, to illustrate the successful composition of Veronese’s “Family of Darius before Alexander”: the frame is split into two triangles, one in dark tones containing figure composition, the other in light ones containing architectural background. Incursions of dark tone into the upper half are matched by incursions of light into the lower, etc. I suspect that Menger, given his own tastes, would have found such normative stipulations concerning aesthetic “success” to be as arbitrary as Birkhoff’s formula for the polygons. One can also wonder what Menger would have made of the closing section of Birkhoff’s book, which was devoted to a discussion of “Creative Art”. Artistic creativity, said Birkhoff, resulted from “highly developed aesthetic judgment [acting] intuitively and rapidly” (p.30), coupled with the “mental energy and interest” necessary for experimentation. There was no need to for any mystical explanation in terms of genius in

21 This was the same Ross who had inspired Roger Fry’s views on aesthetics.

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order to understand the phenomenon. However, in those cases where the theories of an artist “overbalance his aesthetic judgment and experience”, the result was “puzzle-art”. In a clear reference to some kinds of Modernist experimentation, he continued:

“This kind of art has been exemplified in many experiments of recent decades. Any novel artistic form which cannot be appreciated without advance knowledge of the theory underlying it may be suspected of falling into the category of puzzle-art” (p.216).

In light of this, we can understand Menger’s remarks about Birkhoff’s not being an “aesthetic relativist”. All in all, I sense a general coolness in Menger’s recollections of Birkhoff, and I tend to see this encounter as one in which Menger was reminded of his difference as a Modernist -- just as when Clara Wittgenstein baulked before his gift of Masereel. There is more warmth in Menger’s recollections of Birkhoff’s antagonist at MIT, Norbert Wiener, with whom he took long Sunday walks in the countryside outside Cambridge. Menger says that whenever Wiener would begin to voice his criticisms of Birkhoff, he would rebuff him diplomatically and steer the conversation elsewhere. Those discussions included aesthetics, on which Wiener had written the entry in the 1920 Encyclopedia Americana.22 It was an overview of the history of aesthetical theory, from Plato through to Bergson, covering Plotinus, Leibnitz and Tolstoy and others along the way. It also analysed the main theoretical issues present in all aesthetical discussion. The purpose of art, Wiener claimed, was the production of beauty (independent of whether or the artist’s work here was guided by some moral priniciple). What, then, was beauty? The criticisms of subjectivism in aesthetics, Wiener felt, had been misplaced. The idea that beauty depended on the judgment of the beholder did not imply that aesthetic evaluation now became vulnerable to the passing whims of the critic: “An aesthetic judgment which is dependent, let us say, on my feeling of admiration for a certain statue, is no more under my control for that reason than if it were solely dependent on the physical shape of the statue” (p. 201).23 Nor was it possible to link perceived beauty to the “old psychological hedonism which made all the keys of human interest strike one of the twin strings of pleasure and pain” (p. 201). It was probably part of our nature, Wiener suggested, that the claims of sensual pleasure upon us conflicted with the claims of beauty. As to whether beauty was perceived directly or registered emotionally, it was the latter, said Wiener. The same physical primrose, he said, citing lines from Wordsworth, could be regarded with indifference or with joy. What was required was the appropriate emotional response, and once that was possessed, then it could be excited by “any object”: “machine shops or traveling cranes . . . or city slums or barren deserts or in fact

22 See Wiener (1920). The article is not mentioned in Birkhoff (1933). 23 Coincidentally, the exact same claim concerning the independence of subjectively attributed economic value from the free-will of the individual had been made almost fifty years previously by Carl Menger, in the opening chapters of his Grundsätze of 1871.

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anything whatever” (p. 202). Furthermore, there seemed to be no such thing as a particular aesthetic emotion:

“We react to different works of art in very different ways; one excites awe, another pity; we laugh at the comedy and weep at the tragedy, we exhibit the physical symptoms of admiration for this work and those of enjoyment of that. It is impossible for a single physiological reaction to be found that is compatible at once with those of awe, pity, laughter, tears, pleasure, admiration, which is not also found in our reaction to things not beautiful . . . the multifariousness of the aesthetic emotions seems to contradict their possession of any common element” (p. 202)

This was clearly a more liberal view of aesthetic value than Birkhoff’s, and one that was no doubt more attractive to Menger, as he moved through the artworlds of Vienna, The Netherlands and now America.24 Like Boltzmann before him, Menger was greatly impressed by New York and its architecture. In 1930, the city’s first skyscrapers towered above the surrounding buildings, making them even more impressive then than they would become later. American architecture was one of the greatest accomplishments ever, he felt. As for the behaviour of Americans, it evoked mixed feelings. He was struck by the massive waste he observed in daily life, and by what he perceived as Americans’ complete self-assurance in political and economic matters. On the other hand,

“In striking contrast to this posture, I noticed a marked diffidence in most aesthetical matters: hesitant statements of taste and noncommittal judgements about art, music, and literature. Often this attitude was a consequence of ignorance; but even more often I found it paired with knowledge and understanding. Absence of dogmatism is, of course, what empiricists regarded as the only sound way to deal with matters of taste and to discuss questions of aesthetics. But in America of 1930, that frame of mind was conspicuous to a Viennese observer” (Reminiscences, unpublished draft, no page number)

He was also struck by the fact that there seemed to be no recognition in America of artistic and intellectual achievement if it was not accompanied by financial success. “There was no aura of spirituality and no respectability about (sic) the starving poet and musician or the freezing painter living in an attic. If fact, hardly any such people existed” (ibid).

24 Of Wiener’s article, Menger wrote that: “even a strict positivist can subscribe to most of what he said. . . But he is, in my opinion, too vague in saying that according as we apply our taste to an object “in its environment and in its causal chain” or “in complete isolation”, we arrive at moral or aesthetic evaluations” (p. 163). He agreed, however, with Wiener’s affirmation that it was through education and evolution that taste was shaped, and that it was this, rather than any a priori character of the beautiful, that allowed tastes to be shared and art to become a social matter.

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Having spent the first semester of 1931 at the Rice Institute, Texas, where he made a considerable impact both as visitor and teacher, he moved on towards California. As he passed through New Mexico and Arizona, he was deeply impressed by the Native American art of the Southwest:

“I saw in the Indians in New Mexico and Arizona, most of all, their art (in 1930 totally unknown in Europe); the mystical birds and conventionalized beasts on their ceramics; their symbolic wooden dolls and sand paintings; and (in exhibitions) the carvings and totem poles of Northwestern tribes – all this to my taste unrivalled by any works by the traditional pre-abstract American artists of those days that I had seen” (ibid).

Not only would this native artwork later draw Max Ernst and Dorothea Tanning to the Southwest, where they settled in Sedona, but there is now a subtantial literature exploring the varied abstract geometric forms going back millenia in the pottery of the Zuni, Hopi and Anasazi tribes. Politics and the Urge to Abstraction Menger had already begun to speak out against Brouwer’s Intuitionism before leaving for the U.S., in an essay published in 1930, and he did it again in a 1932 public lecture, published a year later.25 In both, the relationship between aesthetics and mathematics was central to his argument and it worked in two distinct ways. The first was his claim that arguments concerning the validity of certain approaches to mathematics could ultimately reflect nothing but the tastes of the mathematician. Brouwer’s appeal to a special ur-intuition as the foundation for the rejection of the law of the excluded middle ultimately carried no greater or lesser authority than Hilbert’s preference for consistency: both were no more than reflections of their respective tastes. As such, they were no more or less valid than, say, a religious belief or a preference for vegetarianism. Such tastes 25 See Menger (1930) and (1933).

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lay outside mathematics itself, belonging to the realm of history and biography. In light of this, said Menger, all that could be required of the mathematician was that he express his tastes clearly, showing the axioms and postulates held at the outset and the rules to be used in combining them to produce theorems. Even if they could not be defended in any ultimate sense, Menger, too, had his own tastes and, not surprisingly, he clearly showed that his were closer to Hilbert’s than Brouwer’s. Justifying this, he once again appealed to aesthetics. He preferred Hilbert’s liberal philosophy insofar as by encouraging the creation of vast and beautiful areas of mathematics, it brought “overwhelming aesthetic advantages”. Even putting aside the question of scientific applications, said Menger, such mathematics was like music, and no-one ever required that music justify itself. Mathematics was art, he was saying, and art for art’s sake. As we move on to the last episode in this exploration of Menger from a Modernist point of view, it is worth reminding ourselves of the troubled personal and political “context” in which all of this is set. From his return to Vienna from Holland in 1927 until sometime in the early 1930’s, Menger was involved in a protracted, priority dispute with Brouwer. This affected their colleagues, with Menger’s mentor, Hans Hahn, acting as intermediary until his death in 1934 when the affair faded away.26 In the realm of politics proper, in 1927, conflict between the national Christian Social government and workers close to the Social Democratic municipal government provoked a riot in which 85 marchers were killed by the police and the Vienna Justizpallas set on fire. In the opening years of the 1930’s, the political tensions mounted, with increased national pressure on the government of Red Vienna, protests by increasingly emboldened Austrian Nazi groups, and the rise to power of Hitler in Germany in 1933. There is a hint in his 1932 talk on Brouwer that Menger was interested in the use of mathematics for the illumination of social relationships, but by late 1933 the tension had reached such a pitch that he abandoned his normal mathematical work and plunged into the consideration of ethics and social relations. The result was his 1934 book, Moral, Wille und Gestaltung (translated decades later as Morality, Decision and Social Organization). Only by fleeing Vienna once again, this time for the Alpine seclusion of Prein, at the foot of the Rax, near Semmering, was he able to finish the work. This book carries us into an unusual dimension of creativity insofar as it was not simply a matheamatical work applied to ethics, but also an attempt by the author to find clarity and solace. If, for Menger, mathematics was art, then surely a creative mathematics aimed at clarifying the troubled social universe of the mathematician himself must be of both aesthetic and therapeutic significance. Although not among the references in Birkhoff’s book on aesthetics, one of the landmark books in the Modernist critical canon is Wilhelm Worringer’s 1908 Abstraction and Empathy. It has been consecrated by students of Modernism not only for the analysis it offers of abstraction in art, but also for the uncanny prescience it showed by appearing when it did, when abstraction was about to flood the European art world.

26 See Van Dalen, Vol. II, pp.643-671.

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Worringer is concerned to show the incompleteness of Theodor Lipps’ Aesthetik which, as we have seen, put empathy at the centre of subjectivist aesthetics. According to Lipps, aesthetic enjoyment required that the subject successfully empathise himself into the object being considered. If the viewer can give himself over to the activity demanded of him by the object “without inward opposition”, then we have positive empathy and, with that, pleasure and aesthetic enjoyment (p.6). Worringer claims that Lipps’ misunderstanding stems from a failure to differentiate between the two forms that our “absolute artistic volition”, our a priori urge to create, may take. By assuming that this volition has been driven by empathy, which seeks gratification in naturalism and in the truths of the organic, Lipps has ignored the “urge to abstraction”: this no-less-important feature of our artistic make-up “finds its beauty in the life-denying inorganic, in the crystalline or, in general terms, in all abstract law and necessity” (p.4). Worringer then proceeds to show how the urge to abstraction has been evident in various cultures across the world throughout history. He does not explicitly refer to Native American art, but he does discuss Oriental art and that of Celtic and other “savage peoples”. Particularly interesting is Worringer’s discussion of what he calls the “psychic presuppositions for the urge to abstraction” (p.15). While his analysis is focused on entire groups or peoples at different periods of history, it is curiously resonant in the context of our account. The explanation of the urge to abstraction is to be found “in these people’s feeling about the world, in their psychic attitude towards the cosmos. Whereas the precondition for the urge to empathy is a happy pantheistic relationship of confidence between man and phenomena of the external world, the urge to abstraction is the outcome of a great inner unrest inspired in man by the phenomena of the outside world; in a religious respect it corresponds to a strongly spiritual dread of space” (p. 15). This spiritual dread is a sense of bewilderment and disconnection, an “instinctive fear conditioned by man’s feeling of being lost in the universe” (p.16). Some cultures, such as Eastern civilizations, acknowledged it fully and did not try to subsume it under cognition.

“Tormented by the entangled inter-relationship and flux of the phenomena of the outer world, such peoples were dominated by an immense need for tranquility. The happiness they sought from art did not consist in the possibility of projecting themselves into the things of the outer world, of enjoying themselves in them, but in the possibility of taking the individual thing of the external world out of its arbitrariness and seeming fortuitousness, of eternalising it by approximation to abstract forms and, in this manner, of finding a point of tranquility and a refuge from appearances” (p.16).

I do not yet know how scholars have adapted Worringer’s blend of cultural and psychological analysis in order to illuminate the mystical and transcendental dimensions of the turn towards abstraction by Mondrian and other Modern artists. I am also aware of the dangers in simply juxtaposing accounts that seem to be unrelated. Yet I am struck by the extent to which Worringer’s account of the urge to abstraction, and the psychic conditions underlying it, captures Menger’s treatment of ethics amidst the chaos of 1933-

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34. Rejecting every existing account of ethics with which he was familar (Kant, Tolstoy, Spinoza especially, and Brentano), Menger systematically excises metaphysics, psychology and history from the field of ethics, stripping it down until it becomes an theoretical static account of imaginary individuals who can hold a limited number of ethical positions.27 The compatibility between such individuals or such sets of ethics can be represented visually. As Menger himself says, his is a geometry of social compatibility.

The value of such abstract compatibility analysis, he says, in both the book and in yet another public lecture in Vienna, is that, in a context of political tumult, it can highlight social possibilities that might not otherwise have been noticed. Similarly, the non-Euclidean geometry of Bolyai and Lobaschevskii was initially an exercise in abstraction that much later turned out to be applicable to the theory of general relativity. While this is in perfect keeping with Gray’s observation that, to the Modernist mathematician, all structures are interesting, it is also difficult to ignore the aesthetic dimensions of this representation of social compatibility and others like it in Menger’s book -- even if they were of course not intended as objects of artistic contemplation. In Worringer’s psychology of art: “The simple line and its development in purely geometrical regularity was bound to offer the greatest possibility of happiness to the man disquieted by the obscurity and entanglement of phenomena. For here the last trace of connection with, and dependence on, life has been effaced, here the highest absolute form, the purest 27 In Peter Gay’s opinion, if there are two irreducible features common to the Modernist effort across the spectrum (literature, visual arts, architecture), they are the elements of “heresy” and “unscrupulous self-criticism”. Beyond the “notes on voluntary associations” that make up the analytical part of Menger’s book, the rest of it is taken up with justifying his radical departure in the interpretation of ethics, some in the form of extended imaginary dialogues with a critic. Incidentally, the only question by the “critic” that Menger chooses not to confront is that of whether his chosen delimitation of the study of ethics is not itself merely a judgment of taste (see, p.35).

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abstraction has been achieved; here is law, here is necessity, while everywhere the caprice of the organic prevails” (op cit, p. 20). Perhaps there is something here that allows us to see Menger’s abstract structures as a source of consolation for him, the aesthetic appeal of line and balance heightened by the knowledge that all had been rigorously constructed. Closing Remarks It is tempting to say that Menger’s Moral, Wille may thus be viewed as a way in which Modernism made its influence felt in social science, but that would be to excessively simplify matters. It was an episode, with few scientific consequences: the book gave rise to no direct literature; it did not mark the establishment of a new approach to mathematical sociology; and it was largely lost in the chaos of the 1930’s and the ensuing war. The richness and fascination of the episode lies in the way in which it illuminates the creative psychology of a mathematician of distinctly Modernist sensibility, as he moved across the fields of geometry, art and sociology. Might it at least be said then that, through its influence upon Oskar Morgenstern, the book indirectly exerted a broader Modernist influence? Here, some nuance is required. Morgenstern was not a fellow Modern for Menger. Neither mathematician nor aesthete, his intellectual and imaginary “universe” was different. There was also an ambivalence, I think, in Morgenstern’s political position and his relationship to the forces of reaction that is difficult to reconcile with anguish or a need for consolation. He did, however, read the book, and felt its relevance to the theoretical quandaries that preoccupied him. He also fell in with von Neumann who himself had also seen the book. While von Neumann deemed Moral, Wille “rather flat”, within a year or two, when political upheaval began to affect him directly, paralysing his work in mathematics, his reaction was quite similar to that of Menger in Vienna several years earlier. He too turned his focus on society, constructing a new mathematics to represent social configurations, the equilibria in which depended upon prevailing social norms, and in order to do so he reached, even more deeply than had Menger, into the Modern mathematics of Hilbert.28 The whole was then presented as a distinctly heretical contribution, like nothing ever seen before, with a good measure of the self-scrutiny and justification required of such an innovation. Like Moral, Wille, the Theory of Games can be regarded an example of the influence of Modernist mathematics upon economics. Alas, with von Neumann being even less an aesthete than his collaborator, the direct connections with the artistic movements of the time are, as far as I can determine, slim to non-existent.29

28 It is not insignificant that by the late 1930’s, not only had Göttingen been destroyed by the Nazis but the Modern mathematics associated with Hilbert had become thoroughly politicized, with Bieberbach condemning it as abstract, Jewish mathematics. On this, see Mehrtens (1987) and, in relation to von Neumann, Leonard (forthcoming). 29 It would be interesting, however, to explore the indirect and symbolic connections between the Theory of Games and the use of the games and mathematics in literature, such as in Hesse’s Glass Bead Game of 1943. Karl Sigmund has considered the signficance of mathematics for the literary work of Musil, Broch and Perutz.

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Gray, Jeremy (2008), Plato’s Ghost, The Modernist Transformation of Mathematics, Princeton University Press. Leonard, Robert (1998), “Ethics and Excluded Middle: Karl Menger and Social Science in Interwar Vienna”, Isis, March. _____________ (forthcoming), Von Neumann, Morgenstern and the Creation of Game Theory, 1900-1960, Cambridge UP. Menger, Karl (1979 [1930]), “On Intuitionism” in Selected Papers, pp. 46-58 ___________ (1979 [1933]), “The New Logic”, in Selected Papers, pp. 18-45 ___________ (1979), Selected Papers on Logic, Didactics, Foundations and Economics, Dordrecht: Reidel ___________ (1994), Reminiscences of the Vienna Circle and the Mathematical Colloquium, edited by Louise Golland, Brian McGuinness and Abe Sklar. Vienna Circle Collection, Volume 20. Dordrecht: Kluwer.

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