Modernism, Fiction and Mathematics

TL;DR

This paper reviews Nina Engelhardt's book exploring the relationship between modernist literature and mathematics, highlighting interdisciplinary connections and thematic insights.

Contribution

It provides an in-depth analysis of how modernist fiction engages with mathematical ideas, offering new perspectives on literary and mathematical intersections.

Findings

Identifies key themes linking modernism and mathematics

Highlights interdisciplinary influence on literature and science

Provides a comprehensive review of Engelhardt's work

Abstract

This is an expanded version of my review of Nina Engelhardt's book "Modernism, Fiction and Mathematics", Edinburgh University Press 2018. A considerably shortened version will appear in the Notices of the AMS.

Full text

Modernism, Fiction and Mathematics

Johann A. Makowsky

Review of:

Nina Engelhardt,

Modernism, Fiction and Mathematics,

Edinburgh Critical Studies in Modernist Culture,

Edinburgh University Press,

Published June 2018 (Hardback), November 2019 (Paperback)

ISBN Paperback: 9781474454841, Hardback: 9781474416238

1. The Book Under Review

Nina Engelhardt’s book is a study of four novels by three authors, Hermann Broch’s trilogy The Sleepwalkers [2, 3], Robert Musil’s The Man without Qualities [21, 23, 24], and Thomas Pynchon’s Gravity’s Rainbow and Against the Day [27, 28].

Her choice of authors and their novels is motivated by the impact the mathematics of the interwar period had on their writing fiction. It is customary in the humanities to describe the cultural ambiance of the interwar period between World War I and World War II as modernism. Hence the title of Engelhardt’s book: Modernism, Mathematics and Fiction. Broch and Musil are indeed modernist authors par excellence. Pynchon is a contemporary American author, usually classified by the literary experts as postmodern.

Engelhard’s book is interesting for the literary minded mathematician for two reasons: First of all it draws attention to three authors who spent a lot of time and thoughts in studying the mathematics of the interwar period and used this experience in the shaping of their respective novels. Broch (1886-1951) and Musil (1880-1942) were Austro-Hungarians and wrote in German. They were translated into English111The first translation of The Man Without Qualities in English was published by Ernst Kaiser and Eithne Wilkins in 1953, 1954 and 1960 immediately after World War II, and are recognized by connoisseurs and experts as the German language counterpart to Joyce and Proust. However, even in the German language world they are not widely read, both because of their length of their novels and the demanding complex narrative. Pynchon is also known for his lengthy novels and vast erudition, and is read mostly by his literary enthusiasts. All three authors deserve the attention of a wider audience among mathematicians and scientists, as I will expound below.

The true merit of Engelhardt’s book lies in its perspective. Without falling into the traps of over-interpretation, she discusses the interplay between modern mathematics and its foundational crisis and the development of the fictional characters described in the novels. She uses the modernist perspective cautiously to put the novels into the wider cultural context of the prewar and interwar period in central Europe, and she avoids successfully making wrong statements about the mathematics of this period.

If some readers of this review pick up only one of these novels and start reading it (possibly but not necessarily together with the relevant chapters of Engelhardt’s book) they will be amply rewarded. Hopefully these readers will then look for the necessary leisure time to go on reading more.

2. Modernism

2.1. Modernity vs Modernism

Modernism as a phenomenon in European culture is traditionally associated with the emancipation of art and science from their exclusive role in representing the true view of the world based on the notion of immediate perception. Kantian epistemology was questioned throughout the 19th century, a cultural pessimism blossomed and the church’s monopoly on the sovereignty of interpretation of the true world was shattered. Its breeding ground is the social changes in the period of intense industrialization before and after WWI.

I do not think that there is a causal connection between the various manifestations of modernism. It is more natural and convenient to see it as a confluence of various developments of social, technological and even political character.

Certainly, some of these developments are interconnected, but not always in an obvious way. The discovery of non-Euclidean geometry definitely was not a result of social or political changes, neither was the emergence of Relativity Theory in physics, nor the discovery of the periodic table in chemistry, nor the discovery of the Mendelian laws in biology. On the other hand, discoveries in the natural sciences and technological progress are interrelated, and there is no doubt that social and political developments were partially accelerated by technological advances.

• •

In painting, the advent of photography, the rise of the industrial bourgeoisie, the changing socio-economic conditions for painters, the confrontation with and the influence of non-European art brought about by colonialism, and the discovery of non-Euclidean geometry, all may have played their role in the development of abstract painting. Painting stopped to be the “true picture” of the world.

• •

In music, the advent of acoustic recording played the same role as photography in painting, and the new discoveries of Helmholtz and others in acoustics played a similar role as the discovery of non-Euclidean geometry. The encounter with non-European music together with the changing social standing of professional musicians opened the ways for experimentation. Twelve-tone music was not only developed by Arnold Schoenberg and his followers in Vienna but also by Nikolai Roslavets in Russia.

• •

In literature the modern and modernist trends are more varied and it is difficult to bring them into a coherent picture. Trends in poetry, drama, and prose are diverse. Influences of philosophy are divergent. Schopenhauer, Nietzsche, Bergson and Kierkegard were among the influential cultural pessimists. Ernst Mach, Carl Stumpf, Edmund Husserl, Ernst Cassirer, the Neo-Kantians, and the philosophers of the Vienna Circle were reexamining the nature of perception and the scientific method, and altered our epistemological outlook.

Peter Szondi’s Theorie des modernen Dramas [31] and Martin Esslin’s The Theater of the Absurd [9], and George Lukacs’s The Theory of the Novel [16, 17] and Milan Kundera’s The Art of the Novel [15], may be viewed as landmarks in the theoretical understanding of modern drama and modern novels.

For our discussion here it is noteworthy that among the German language authors of novels Hermann Broch, Hermann Hesse, Thomas Mann, and Robert Musil (in alphabetical order) are considered paradigmatic for modernist prose in German. The same holds for James Joyce in English and Marcel Proust and Paul Valéry in French.

• •

Modern mathematics is largely the result of two internal developments in mathematics: the discovery of non-Euclidean geometry and the emergence of set theory. In the years between 1874 and 1897 Georg Cantor created set theory single-handed, followed by Felix Hausdorff’s Grundzüge der Mengenlehre [13] and in 1899 Hilbert published his Foundations of Geometry [14] and started emphasizing the use of axiomatic methods, see [7]. To the readers of the NOTICES, “modernism in mathematics” may sound familiar from Yuri Manin’s review [19] of Jeremy Gray’s Plato’s Ghost: The Modernist Transformation of Mathematics [12], and from Frank Quinn’s Invisible Revolution of Modern Mathematics [29]. In his review of [12] Solomon Feferman [10] describes “modern mathematics” as follows:

Modern mathematics–in the sense the term is used by working mathematicians these days–took shape in the period from 1890 to 1930, mainly in Germany and France. Strikingly new concepts were introduced, new methods were employed, and whole new areas of specialization emerged, while other themes were relegated to the dusty shelves of history. At the same time, the nature of mathematical truth and even the consistency of mathematics were put into question, as mathematicians, logicians and philosophers grappled with the subject’s very foundations. […] modernism is defined as 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 naive relationship with the day-to-day world.

Books reflecting the spirit of modern mathematics are sometimes called “Modern Algebra” (van der Waerden), “Modern Analysis” (Dieudonné, Rudin) and the modern approach culminates in Bourbaki’s unfinished multi-volume enterprise.

• •

In the realm of classical Number Theory one might be tempted to consider the Prime Number Theorem of Hadamard and de La Vallée-Poussin as the first success of modern mathematics for its use of analytic functions in the proof. Hardy’s belief that no “elementary proof” of it should exists reflects his belief that modern methods give rise to deeper mathematical understanding and can not be dispensed of. Hélas, Erdős and Selberg shattered this belief. For a history of the Prime Number Theorem see [11]. Similarly, Wiles’ proof of the Shimura conjectures is just one of the later achievements of modern mathematics, leading to a proof of Fermat’s Last Theorem. Its original proof uses Grothendieck’s language of schemes and universes. To what extent this can be eliminated is discussed by A. MacIntyre [18] and McLarty [20]. The question to what extent Fermat’s Last Theorem admits an “elementary proof” is still open.

In order to appreciate Nina Engelhardt’s book we can summarize the modernist aspect of (modern) mathematics as follows: The intuitive notion of mathematical truth (facts) is replaced by the axiomatic description of relationships where the only concern is consistency. Mathematics passes from describing the true world to describing mere possibilities.

3. Formalists vs intuitionists

One of the first modern(ist) mathematicians who was also a successful philosopher, playwright and essayist was Felix Hausdorff. He started his career as a successful writer under the name of Paul Mongré, and his writings could easily be classified as modernist. In 1903/04, in his lecture course Time and Space he remarked about mathematics in general, reflecting on Hilbert’s view on the axiomatic method:

Mathematics stands completely apart not only from the actual meaning that one attributes to its concepts but also from the actual validity one ascribes to its propositions. Its undefinable concepts are arbitrarily chosen objects of thought, its axioms are also arbitrary, though chosen so as to be free from contradiction. Mathematics is the science of pure thought, just as formal logic. 222 Quoted from [25]

His literary publications were mostly published between 1897 and 1904 and consisted of a volume of poems, a play, a book on epistemology, and a volume of aphorisms. He wrote no novel, and as a philosopher he was mostly influenced by the younger Friedrich Nietzsche. However, he was in correspondence with Moritz Schlick, the founder of the Vienna Circle, and seemingly influenced Schlick’s early views on time and space.

The modern(ist) view of mathematics led to heated debates in continental Europe between Formalists and Intuitionists, with heavy political overtones. In philosophical circles such as the Vienna Circle around Hans Hahn and Moritz Schlick, and its Berlin version around Hans Reichenbach this view led to a new and very strict definition of what constitutes the scientific method. The two philosophical circles were also influenced by Ludwig Wittgenstein and the emergence of mathematical logic. Both Kurt Gödel and Karl Menger were students of Hans Hahn. Among the eminent mathematicians Karl Menger was an active member of the Vienna Circle and David Hilbert became a member of the Berlin Circle.

4. Modernist writers and mathematics

While I studied mathematics in the late 1960s I was preoccupied by the question of the nature of mathematical thought. I read a lot of material pertaining to this question: philosophy, psychology, neurology, and also literary works. Among the latter I came across five authors who struck me as significantly influenced by modern mathematical and/or musical thought, Paul Valéry, Robert Musil, Hermann Broch and Hermann Hesse, and to a lesser extent Thomas Mann. All of them are Europeans who wrote mostly before WWII. Broch and Musil are the subjects of Engelhardt’s book. Valéry, Mann and Hesse are not. I will briefly discuss, whether they should have been. Thomas Pynchon is American and contemporary and writes in English. Engelhardt’s book originated as a Ph.D. thesis from 2012 in the department of English Literature at the University of Edinburgh, Mathematics in literature: modernist interrelations in novels by Thomas Pynchon, Hermann Broch, and Robert Musil. Pynchon’s two novels both deal with the mathematics and science, and are set partially in continental Europe of the period of Broch and Musil. I will also briefly discuss why Pynchon, besides fitting a thesis in the department of English Literature, is a major author discussed in Engelhardt’s book.

Paul Valéry, poet and philosopher

Paul Valéry, a symbolist/modernist French man of letters is best known for his poetry. However, he was also interested in philosophy and mathematics, and was supposed to write a preface for F. Le Lionnais’ Les grands courants de la pensée mathématique, but died in 1945 before writing his preface. Raymond Poincaré, Louis de Broglie, André Gide, Henri Bergson, and Albert Einstein all appreciated Valéry’s thinking. They became friends personally and by correspondence. His L’idée fixe ou deux hommes à la mer [32, 33], written in 1932, is a brilliant reflection on the relationship between art and science, where he describes the gap between experience and theory as so big that only a visionary architect can possible bridge it. The text resonates Valéry’s experience from attending a lecture by Albert Einstein on Time, Space and Relativity.

Hermann Broch and the Vienna Circle

Hermann Broch was born in 1886 in Vienna to a prosperous Jewish family and worked for some time in his family’s factory. As the oldest son, he was expected to take over his father’s textile factory, therefore, he attended a technical college for textile manufacture and a spinning and weaving college. Although he had from early on philosophical, mathematical and literary interests, he pursued them only privately, mostly at night, while managing the textile factury during the day. In the years 1904-1905 he attended seminars and public lectures in mathematics and philosophy at Vienna University, especially those given by Ludwig Boltzmann.

After an unsuccessful marriage which led to his divorce in 1923, he sold the textile factory in 1927 to devote himself full time to his intellectual interests. He studied mathematics, philosophy, physics and also psychology and cultural history at the University of Vienna. Philosophically he was, like Hausdorff, first under the influence of Schopenhauer, Nietzsche and Weininger, but later turned to neo-Kantian philosophy and logicism and got involved with the Vienna Circle, siding with Wittgenstein and Menger rather than being a strict positivist. From 1913 on he started to publish philosophical essays.

Two recent theses which appeared during or after Engelhardt’s work on her book are devoted in great detail to Broch’s philosophical and scientific background and its impact on his literary work: Ludwig Wittgenstein and Hermann Broch: The Need for Fiction and Logic in Moral Philosophy by C. W. Bailes [1] and Zweige eines einzigen Stamms. Hermann Broch’s Reflections on Science and Literature by H.A.M. Brookhuis [6]. Broch studied at Vienna University until 1930 having among his teachers Moritz Schlick, Rudolf Carnap, Hans Hahn and Karl Menger.

Originally he was more interested in a scientific career and wanted to write a Ph.D., but he became discouraged, due his lack of knowledge in Latin, which was, at that time, a compulsory subject for getting a Ph.D. The goal of writing a thesis in mathematics was delayed indefinitely and finally abandoned also because of his disenchantment with science due to its increased specialization. It is during this period that he wrote his trilogy The Sleepwalkers published in 1931 which marked the beginning of his career as a novelist, and which became known as a modernist novel par excellence. He also wrote a novel, Die Unbekannte Grösse [4, 5] inspired by the events around Einstein’s theory and its experimental verification during an eclipse visible only in the southern hemisphere. Among the writers which N. Engelhardt discusses, H. Broch is definitely the most educated mathematician.

Robert Musil and the Berlin Circle

Robert Musil was born in 1880 in Klagenfurt (Austria) into a middle class family. His father was a professor of mechanical engineering at the German Technical University in Brünn (now Brno, Czech Republic), and, later, rose to hereditary nobility in the Austro-Hungarian Empire. Musil held two doctorates, one in mechanical engineering at his father’s university (1901), and one in in psychology and philosophy at the University of Berlin under the renowned Professor Carl Stumpf. Carl Stumpf wrote his Habilitationsschrift in 1870 in Göttingen on the foundations of mathematics. It was recently republished [30].

M. Schlick’s early philosophy shows both appreciation and criticism of Stumpf’s epistemology. Among Stumpf’s students we also find Edmund Husserl, who established the school of phenomenology, and who had studied mathematics with Kronecker and Weierstrass. Musil’s majors in Berlin were “logics and experimental psychology”, and he was examined in 1908 in philosophy, natural science and mathematics. A. Schwarz of the renowned Schwarz’ inequality was among his doctoral examiners. The topic of his thesis was Beiträge zur Beurteilung der Lehren Machs. His thesis and other technical writings, covering the period from 1904-1922 were republished in 1980 as [22]. In 1906 his first novel was published and at the same time he was developing an apparatus to research how people experience color. From then on he dedicated his time exclusively to his literary career, interrupted only by his military service during WWI. From 1918 till 1931 Musil lives in Vienna. We do not know whether Musil had direct contact with the Vienna Circle during these years in Vienna. In 1931-33 Musil lived in Berlin and was an active participant in the Berlin Circle around Hans Reichenbach and Richard von Mises. However, the first two volumes of The Man Without Qualities were already published in 1930. Musil’s awareness of modern mathematics and the foundations of geometry must have been acquired while he studied for his doctorate in Berlin. Musil’s background in engineering, psychology and mathematics most definitely influenced his writing not only in The Man without Qualities. In his 1906 novel The Confusions of Young Törless the confusions are manifold, ranging from the sexual to the foundations of geometry.

Thomas Pynchon’s failed attempt to get a Ph.D. in mathematics

Thomas Pynchon is famously reclusive. About his mathematical and philosophical education we know rather little. As a writer he belongs more to post-modernism than to modernism.

I gather from the Wikipedia article about him the following: He was born in 1937 and holds an English degree from Cornell. Starting in 1953 he studied engineering physics at Cornell University, but left at the end of his second year to serve in the U.S. Navy. In 1957, he returned to Cornell to pursue a degree in English. From February 1960 to September 1962, he was employed as a technical writer at Boeing in Seattle. In 1964, his application to study mathematics as a graduate student at the University of California, Berkeley was turned down. The two novels discussed in Engelhardt’s book are Gravity’s Rainbow, published in 1963, and Against the Day, published in 2006. The former draws from his experience at Boeing, the latter contains a lot of mathematical material pertaining to Sofia Kovalevskaya and to Hilbert’s school in Göttingen. Pynchon seemingly researched this material with the help of Michael Naumann, German secretary of culture from 1998 until 2001. The only other good source for Pynchon’s biographical material is his lengthy autobiographical introduction to the collection of short stories Slow Learner [26], published in 1984.

The novels of Pynchon are considered difficult to read. Nevertheless, he has his fans who created a Wikipedia-like website (https://pynchonwiki.com/), which should help potential readers to enter Pynchon’s world. Literary critiques celebrate him as one of the most important American novelists of the second half of the 20th century. Gravity’s Rainbow and, even more so, Against the Day are novels fitting the topic of Engelhardt’s study. Contrasting him to Broch and Musil, comparing modernist to post-modernist writing, makes Engelhardt’s book a gem among the few studies linking mathematics and literature.

Hermann Hesse and Thomas Mann

There are two German authors who come to my mind for their modernist novels dealing with formal aspects of mathematics and music: Thomas Mann with his Doktor Faustus and Hermann Hesse with his Magister Ludi (aka The Glass Bead Game). Thomas Mann’s Doktor Faustus does not deal with Mathematics and its foundational crisis but instead with music and the atonality of the Vienna School. Thomas Mann was not an expert in modern music. He commissioned Theodor Adorno to help with the musical aspects of his novel. But in a a very precise sense his novel could have been discussed in the Framework of Modernism, Music and Fiction. It was wise of Nina Engelhardt not to include Thomas Mann’s novel in her discussion, but it may be a tempting topic for further research.

In Hermann Hesse’s novel the situation is more complex. One of the central themes of the novel is the Glass Bead Game and the way it is performed in public by its masters. The details of the game itself are only vaguely described in the novel as an amalgam of mathematics, music and the Chinese I Ching. But it says that playing the game well requires years of hard study of music, mathematics, and cultural history. The game is essentially an abstract synthesis of all arts and sciences. It proceeds by players making deep, and sometimes unexpected, connections between seemingly unrelated topics. The novel is a dystopia in a post-historic setting. The mathematics is not specified in detail, but the sources used by Hesse concerning music, history, and Indian and Chinese culture are well documented and identified.

Carl Jacob Burckhardt, the Swiss historian and diplomat, of the same family as the eminent historian and scholar of Renaissance culture Jacob Burckhardt, was a close friend of Hesse and can be recognized as one of the novel’s protagonists. His career alternated between periods of academic historical research and diplomatic postings; the most prominent of the latter were League of Nations High Commissioner for the Free City of Danzig (1937–39) and President of the International Committee of the Red Cross (1945–48). […] He was teaching contemporary history at the University of Zurich from 1927 until 1932333 Quoted from the wikipedia https://en.wikipedia.org/wiki/Carl_Jacob_Burckhardt.

In the Hesse archive there is no written evidence of Hesse’s inspiration concerning modern mathematics. However, in a talk given by Hermann Weyl, [34] we learn, that Hermann Weyl, Hermann Hesse and Carl Jacob Burckhardt were close friends.

Commenting on his decision to leave Zurich to succeed Hilbert in Göttingen in 1930, Weyl writes:

[…] it could have possibly been right to stay in Zurich, where my position was specially favorable to a vita contemplativa, paired with, relatively speaking, modest external impact. It was not easy for me to come to a clear decision. I have pondered the question in my mind together with two men, with Jacob Burckhardt and Hermann Hesse. The fate of the latter is particularly close to me. He went “South”, not only geographically. His example is tempting, but also a warning. For he achieved only a deeper loneliness and a mild desperation, which strangely permeated the happiness of his senses, his eyes and his writing, which was at the same time exhilarating and loosening up. Burckhardt on the other hand rejected the offer to succeed Ranke in Berlin with self-evident determination.

Hermann Weyl was known for his inspiring lectures, celebrating mathematics as a performing art. It is then likely, that he was both the model for the Magister Ludi celebrating the glass bead game, and the source for Hesse’s awareness of modern mathematics as the art of the possible, rather than the science of true nature. The impact of Hermann Weyl on Hesse’s Magister Ludi has not been explored in depth, but many mathematicians reading the novel conjecture modern formalist mathematics to be a major source for Hesse’s concept of the Glass Bead Game. Few know that the source of Hesse’s knowledge seems to be Hermann Weyl.

5. What Engelhardt achieves

The book under review has no abstract, but the underlying Ph.D. thesis [8], from 2012 does. It says:

The focus of this thesis is on four novels’ illustrations of the parallels and interrelations between the foundational crisis of mathematics and the political, linguistic, and epistemological crises around the turn to the twentieth century. While the latter crises with their climax in the First World War are commonly agreed to define modern culture and literature, this thesis concentrates on their relations with the “modernist transformation” of mathematics as illustrated in Thomas Pynchon’s Against the Day (2006) and Gravity’s Rainbow (1973), Hermann Broch’s The Sleepwalkers (1930-1932), and Robert Musil’s The Man without Qualities (1930/32).

I have explained why the choice of these four novels is justified by examining the mathematical background of the three authors Broch, Musil and Pynchon. I have also indicated that, in a further study along the same lines, a close examination of Hesse’s Glass Bead Game could be a valuable addition.

Engelhardt’s view of the “modernist transformation” of mathematics is close to what Quinn describes in [29], although slightly less precise. In her words the impact of this “modernist transformation” on fiction consists in the development of autonomy and independence from the natural world, almost a kind of escapism into possible worlds. Musil’s man without qualities is a man of all possibilities.

In the abstract published by Edinburgh University Press advertising the book on their website, it says further:

[The four novels] accord mathematics and its modernist transformation a central place in their visions and present it as interrelated with political, linguistic, epistemological and ethical developments in the modern West. Not least, the texts explore the freedoms and opportunities that the mathematical crisis implies and relate the emerging notion of ‘fictional’ characteristics of mathematics to the possibilities of literature. By exploring how the novels accord mathematics a central role as a particularly telling indicator of modernist transformations, this book argues that imaginative works contribute to establishing mathematics as part of modernist culture.

Engelhardt’s monograph offers the readers a new framework of textual and cultural analysis in which one can understand the modernist and postmodern interplay of literature and mathematics. It is fair to say that in her book Nina Engelhardt does succeed in giving us an inspiring tour d’horizon of this interplay.

Souces and copyright of the illustrations

In order of their appearence.

• Hermann Broch:

public domain

https://en.wikipedia.org/wiki/Hermann_Broch

• Robert Musil:

public domain

https://en.wikipedia.org/wiki/Robert_Musil

• Thomas Pynchon:

public domain

https://www.vulture.com/2013/08/thomas-pynchon-bleeding-edge.html

Credit: Pynchon, age 16, in his 1953 high-school yearbook, one of the few known photos of the author. Photo: Getty

https://www.cs.mcgill.ca/ rwest/wikispeedia/wpcd/images/281/28137.jpg.htm

• Paul Valery:

public domain

https://en.wikipedia.org/wiki/Paul_Valéry

• Hans Hahn:

public domain

https://geschichte.univie.ac.at/en/persons/hans-hahn-prof-dr

Hans Hahn (1879-1934), Mathematics Courtesy: Archive of the University of Vienna, picture archive Originator: Theo Bauer, Wien XIV. Signatur: 106.I.1210 1927

• Moritz Schlick:

public domain

https://en.wikipedia.org/wiki/Moritz_Schlick

• Karl Menger:

public Domain

https://en.wikipedia.org/wiki/Karl_Menger

Photograph of mathematician Karl Menger, from the January 1970 issue of the Shimer College Bulletin. Shimer College -

https://archive.org/details/ShimerCollegeBulletin_1970-01 , page 1

• Carl Stumpf:

public domain

https://en.wikipedia.org/wiki/Carl_Stumpf

• Hermann Hesse:

public domain

https://en.wikipedia.org/wiki/Hermann_Hesse#/media/File:Hermann_Hesse.jpg

1905 portrait by Ernst Würtenberger (1868–1934)

• Carl Jacob Burckhardt:

public domain

https://en.wikipedia.org/wiki/Carl_Jacob_Burckhardt

Copyright: http://www.redcross.int/EN/copyright.html

Any part of this website can be cited, copied, translated into other languages or adapted to meet local needs without prior permission from any component of the Movement, provided that the source is clearly stated.

• Hermann Weyl:

public domain

https://en.wikipedia.org/wiki/Hermann_Weyl

Source ETH-Bibliothek Zürich, Bildarchiv, CC BY-SA 3.0

https://en.wikipedia.org/wiki/Hermann_Weyl#/media/File:Hermann_Weyl_ETH-Bib_Portr_00890.jpg