“A Proteus who transforms himself ceaselessly in order to elude the grip of his adversary, not becoming himself again until after the final victory.” Thus Hermann Weyl (1885–1955) appeared to his eminent younger colleagues Claude Chevalley and André Weil. Surprising words to describe a mathematician, but apt for the amazing variety of shapes and forms in which Weyl’s extraordinary abilities revealed themselves, for “among all the mathematicians who began their working life in the twentieth century, Herman Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universal mathematicians of the nineteenth century, Hilbert and Poincaré,” in the view of Freeman Dyson. “He was indeed not only a great mathematician but a great mathematical writer,” wrote another colleague.1 This anthology presents a spectrum of Weyl’s later mathematical writings, which together give a portrait of the man and the mathematician. The works included have been chosen for their accessibility, but they do also include, where needed, the mathematical details that give vivid specificity to his account. Weyl’s essays will convey pleasure and profit both to general readers, curious to learn directly from a master mathematician, as well as to those more versed, who want to study his unique vision. Those who wish to explore further his seminal work in physics in its philosophic and mathematical contexts will find relevant writings in a companion anthology, Mind and Nature [2009a], which gathers his writings in these fields; as a result of Weyl’s inclusive vision, these two anthologies overlap and complement each other at many points.
The protean Weyl drew his many shapes from rich and complex life-experience, in which mathematics formed only one strand in a complex tapestry. Already philosophic in temperament as a teenager who pored over Kant, Weyl was deeply formed by his work with David Hilbert, his teacher and mentor at the University of Göttingen. Weyl recalled being “a country lad of eighteen” who became entranced by Hilbert, a Pied Piper “seducing so many rats to follow him into the deep river of mathematics.”2 Widely considered to have been Hilbert’s favorite student, Weyl graduated in 1910 and stayed on as a Privatdozent until 1913, when he took a post at the Eidgenössiche Technische Hochschule (ETH at Zürich.3
There he met Albert Einstein, who during 1913–1914 was in the midst of his struggle to generalize relativity theory, which led him to study Bernhard Riemann’s sophisticated mathematics of curved higher-dimensional manifolds. Returning to Switzerland in 1916 after his military service, Weyl’s “mathematical mind was as blank as any veteran’s,” but Einstein’s general relativity paper of that year “set me afire.” Weyl went on to write his “symphonic masterpiece” (as Einstein called it), Space-Time-Matter [1918b], one of the first and perhaps still the greatest exposition of relativity. As part of his explanatory work, Weyl went on to formulate his seminal gauge theory unifying electromagnetism and gravitation [1918c], which Einstein hailed as “a first-class stroke of genius.”4 Though the initial hopes for this theory faded in the light of difficulties noted by Einstein (and later the need to incorporate quantum theory), Weyl continued to ponder the generalizations and implications of his 1918 theory. Looking back, Weyl’s work laid the foundation of the gauge theories that, fifty years later, unified the strong, weak, and electromagnetic theories, fulfilling his initial aspiration in ways he had not dreamed.
Weyl eventually succeeded Hilbert at Göttingen, though only after protracted hesitation: first offered a professorship there in 1923 (to replace Felix Klein, another great mathematician important to his development), Weyl refused, happier to remain in Zürich, where he could find relief for his asthma in the mountains. He finally accepted the call to Göttingen in 1930, when Hilbert himself retired. The first essay in this volume dates from that year, as the newly-appointed Weyl travelled to Jena to speak to mathematics students. This essay, “Levels of Infinity,” was not included in Weyl’s collected papers and has not been reprinted since 1931 nor ever before translated into English.5 It gives an arresting portrait of his intellectual development and his struggle with basic questions in the foundations of mathematics.
Though devoted to Hilbert, Weyl for a time sided against him in an ongoing controversy that continues to simmer under the surface of mathematical practice. In the course of introducing unheard-of levels and degrees of infinity during the 1880s, Georg Cantor opened new mathematical realms that excited but also disturbed thoughtful practitioners.6 Against Cantor, his eminent contemporary Leopold Kronecker argued that only the integers had fully authentic mathematical reality, challenging innovators to express their new insights in terms of this mathematical bedrock or else abandon them as unsound, following his motto “God created the integers; everything else is the work of man.”7 L.E.J. Brouwer, a distinguished topologist four years older than Weyl, made this cause his own, under the banner of “intuitionism”: mathematics should return to its roots in human intuition, radically restricting its use of infinite quantities to those that could be grounded in possible acts of intuition. This would mean the exclusion of speculatively constituted sets, such as the set of all sets of real numbers, for which no intuition could possibly exist.8
To that end, Brouwer disallowed the use of the “principle of the excluded middle”— that any proposition either is true or its negation is true — especially when applied to infinite sets. This principle could be applied to argue that, if the negation of some mathematical proposition is demonstrably false, therefore the proposition must be true merely by elimination, without providing a positive demonstration. Brouwer advocated abandoning such propositions as not proven. For instance, let us examine the proposition “there exist two irrational quantities a and b such that ab is rational.” The Pythagoreans had already shown that 
is irrational. Consider the quantity 
; by the principle of the excluded middle, it is either rational or irrational. If it is rational, the proof is complete, for it exemplifies the quantity ab sought in the case 
. If, on the contrary, 
is irrational, then set 
, so that
which is rational, thus implying that the proposition is true. From Brouwer’s point of view, though, the assumption that 
is rational must be supported by positive proof, not just used as a hypothesis in a proof by exclusion.9 Merely sidestepping the issue by moving the symbols around does not address the underlying question: what sort of existence does 
have?
This critique had deep implications for the whole conduct of mathematics. For instance, Cantor had conjectured that no cardinal number lies between those denoting the countability of the integers, 
0, and the uncountable continuum of real numbers, c = 1. Brouwer considered this “continuum hypothesis” meaningless because the idea of “numbering” the continuum is an empty, abstract construct that lacks any intuitive substance, compared to our intuition of the integers. Most of set theory and large parts of analysis would have to be vacated, on Brouwer’s view. Ludwig Wittgenstein was also among those who considered that set theory was “pernicious” and “wrong” because it leads to “utter nonsense” in mathematics.10
Weyl was deeply struck by Brouwer’s arguments and addressed the problematic groundwork of mathematics in his seminal book The Continuum (1918), avoiding the most problematic recourse to nondenumerable infinities through a redefinition of the concept of number in terms of denumerable Cauchy sequences; Weyl will return to this idea in several of the papers in this volume. Weyl’s work on the mathematical continuum is notably simultaneous with his daring attempt to generalize the space-time continuum through his invention of gauge theories; a preoccupation with seamless continuity versus atomism runs through both streams of his thought.11
In the early 1920s, Weyl went so far as to set his own foundational approach aside and acclaim Brouwer as “die Revolution.”12 Hilbert, in contrast, advocated a purely abstract formalism in which mathematics became a meaningless game played with symbols, utterly detached from intuition and hence untainted by human fallacies and illusions. Hilbert thought thereby to assure at least the noncontradictoriness of mathematics, leaving for the future to prove its consistency through iron-clad logical means that would owe nothing to mere intuition.13 For him, Weyl’s metamorphosis into an acolyte of intuitionism verged on betrayal of what Hilbert thought was the essential mathematical project, which included the “paradise” (as Hilbert called it) of Cantor’s transfinite numbers. But by the mid-1920s, Weyl’s initial enthusiasm for intuitionism had given way to a more measured view of “the revolution,” which he (along with Hilbert) judged would leave in ruins too much beautiful and important mathematics that could not be proved using intuitionistically pure arguments.
“Levels of Infinity” gives an important portrait of Weyl’s views in the wake of this critical period. Now speaking as Hilbert’s successor at Göttingen, Weyl aims to give a larger, more inclusive view that will do justice both to the core of intuitionistic insight, in which he still believes, and yet take into account Hilbert’s formalism. From the first line, Weyl embraces mathematics as “the science of the infinite” but at the same time emphasizes his sense of “the impossibility of grasping the continuum as a fixed being.” Weyl sets forth a series of levels of infinity, giving “an ordered manifold of possibilities producible according to a fixed procedure and open to infinity,” an openness he eloquently celebrates as the essence of mathematical — and human — freedom: “Mathematics is not the stiff and paralyzing schema the layman prefers to imagine; rather, with mathematics we stand precisely at that intersection of bondage and freedom that is the essence of the human itself.”14
His critique of the actual, finished infinite leads Weyl to advocate explicit construction of mathematical entities as the antidote to the “mathematically empty” strategem of considering existent whatever is not categorically impossible: “Only by means of this constructive turn is a mathematical mastery, an analysis, of continuity possible.” Weyl’s philosophic seriousness cannot brook the triviality he finds in Hilbert’s depiction of mathematics as a meaningless game with empty symbols. Instead, Weyl emphasizes the profound implications of the symbolic mode of constructive mathematics, which he connects with the pervasive philosophic implications of symbolic forms throughout human thought and expression.15 Though strongly influenced by his contact with the phenomenological philosopher Edmund Husserl, Weyl’s sense of the priority of symbolic expression goes beyond what is explicable merely through the phenomenal flux of experience: “We cannot deny that there lives in us a need for theory that is absolutely inexplicable from a merely phenomenalist standpoint. That need has a drive to create directed to a symbolic shaping of the transcendent, which demands satisfaction . . . . All creative shaping by humans receives its deep holiness and dignity from this connection.” Though he disclaimed theism as well as atheism, Weyl read attentively the writings of Meister Eckhart and thought deeply about the connection between theology and mathematics, as when he describes “God as the completed infinite, [which] can neither break into the human through revelation nor can the human by mystic contemplation break through to Him. We can only represent the completed infinite in the symbol.”16
In 1931, Weyl spoke on “Topology and Abstract Algebra as Two Roads of Mathematical Comprehension” at a summer course for Swiss teachers, showing his interest in improving secondary instruction and guiding its development. He begins by invoking his teacher Felix Klein, who had been much involved in this cause.17 The problem of intuition is foremost in Weyl’s mind; he notes that we do not merely want to follow formal reasonings blindly but “want to understand the idea of the proof, the deeper context,” not just its “machinery.” Weyl praises Klein’s “intuitive perception” and brings forward topology as exemplifying this kind of visually compelling approach to mathematics. At the same time, though, Weyl notes how the algebraic path, though less intuitively appealing, completes and makes rigorous what topology had discerned more generally. His image of two Wege, paths or roads, evokes the sense of their different, even contradictory, qualities, though in many of his examples he wishes to stress that these two ways ultimately complement each other. His conclusion, though, sides with topology as providing the essential initial insight into what he calls the “mathematical substance” that only afterwards is fruitfully mined by algebraic labor. The image of mining also lies behind his rather dark ending, which speaks of the exhaustion of the abstract vein and the hard times he foresees for mathematicians.18
In 1939, he phrased this alternative much more starkly: “In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.” Those were not the only angels and devils struggling at the time. The intervening dark years included the tragic ending of the great period of Göttingen mathematics, which Weyl experienced first-hand. He had returned there reluctantly, having been happy since 1913 in Zürich, where he felt greater freedom than he had in Germany. His hesitation was prescient; the growing darkness of the early 1930s extinguished the brilliance of Göttingen, whose Jewish mathematicians were systematically stripped of their offices and attacked by the Nazi racial laws. By 1933, Weyl, with his Jewish wife and liberal sympathies, left for the newly-founded Institute for Advanced Studies at Princeton, where his colleagues included Einstein, Kurt Gödel, and John von Neumann.19
This dark history shadows Weyl’s portrait of Emmy Noether (1882–1935), his contemporary, friend, and mathematical colleague, who also had fled to the United States, though she only lived two years after taking up a position at Bryn Mawr. Delivered as a memorial address after her early death, Weyl brings to life an extraordinary figure he praises as mathematician who struggled against prejudice, yet without becoming embittered. From our present distance in time, we may be disturbed by the categorization of her as a “woman mathematician” implicit in Albert Einstein’s description of her as “the most significant creative mathematical genius thus far produced since the higher education of women began” or Weyl’s description of Noether as “a great mathematician, the greatest, I firmly believe, her sex has ever produced, and a great woman.”20 Rather than being condescending, Einstein and Weyl were trying to do justice to her situation, and that of women scientists in general, in light of the historical context and the prevalent prejudices against them. Hilbert himself had stood up for her: “I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, the [Göttingen Academic] Senate is not a bath-house.”21 But even Hilbert’s strong advocacy only resulted in an irregular, unpaid appointment for her.
Though Weyl shows the injustices she endured, his eulogy gives a detailed account of her mathematical accomplishments, as well as her personality and unique life-trajectory. Here, too, he takes on the mantle of Hilbert as Noether’s champion, now laboring not to win her the kind of university post to which her merits entitled her but to give her the posthumous recognition she now enjoys. Thanks to Noether, and to Hilbert, Einstein, and Weyl, now those passionate about science and mathematics can pursue it in light of her example, the heritage of prejudice increasingly relegated to the all-too-human past.
Weyl’s account of Noether situates her in the vortex of the eventful history of modern mathematics, her insights forming part of complex developments that go far beyond the “merely-personal” aspects of gender, religion, or nationality. Physicists may know best Noether’s principle connecting the symmetries of dynamics with conserved quantities (as symmetry in time-displacement is manifest in conservation of energy, for instance), which Weyl embeds in a much larger picture of her work. His overview will help contemporary readers gain a fuller picture of her accomplishments, which have deep implications for mathematics and physics both. Weyl emphasizes her wide range, which included heroic formal computations and conceptual axiomatic thinking; as such, she exemplifies his recurrent theme of the complementary interdependence of axiomatic and constructive approaches to mathematics. “She originated above all a new and epoch-making style of thinking in algebra,” the generalized modern approach, which Weyl above contrasted with the intuitive sweep of topology. An influential teacher, Noether’s students and followers carried her influence into the ensuing developments in mathematics and its increasingly “abstract” style.22
Weyl’s eulogy of Noether was written in English, as were the remainder of the works in this anthology, befitting his new life in the United States. Here and at many other points of his subsequent English writings, Weyl revels in the idioms and possibilities of his new tongue; while still remaining very characteristic of his personal style, his English writings are notably clearer than his often intricate, difficult German prose. His ability to adapt to a new land and a new language, indeed to revel in his rejuvenation and new life, were not the least of his protean metamorphoses.
In his new American phase, Weyl became a significant figure in public intellectual life, though never very well-known or widely popular; he neither received nor craved the réclame that dogged Einstein. He gave celebratory addresses on a number of occasions, including the bicentennial of the University of Pennsylvania (1940), for which he delivered “The Mathematical Way of Thinking.”23 Weyl emphasizes the larger significance of mathematics for the sciences and everyday thinking, even above its internal workings, left to itself. Where Brouwer was cool to applied mathematics, Weyl had long been deeply interested in physics and the broadest implications of mathematics. Consistent with his own practice, he refuses to divide truth “into watertight compartments like historic, philosophical, mathematical thinking, etc. We mathematicians are no Ku Klux Klan with a secret ritual of thinking,” using a vivid allusion to his new American milieu to make his larger point. Mathematics, like democracy, is a “way of life,” not a lifeless set of formal axioms nor purely introspective intuitions, à la Brouwer.24
Weyl critiques the common description of mathematics as “abstract,” breathing thin air and reducing everything to thinner outlines. On the contrary, mathematics requires the man in the street “to look things much more squarely in the face; his belief in words must be shattered; he must learn to think more concretely.” The full potency of “purely symbolic construction” lies behind what we unthinkingly call “abstraction.” Weyl’s charming example of the exact height of Longs Peak shows that the “abstract” concept of potential is, in fact, more concrete than the common notion of altitude. Mindful of “the witchcraft of words in the political sphere,” Weyl argues that “the scientist must thrust through the fog of abstract words to reach to concrete rock of reality.” To do so, “the intuitive picture must be exchanged for a symbolic construction,” reminding us also of Weyl’s own reconsiderations of the role of intuition in mathematics, by comparison with the power of symbols.25
His ensuing examples include relativity theory, basic concepts of number, and topology, whose natural connections and analogies are foremost in his mind, rather than any rigid separation between “pure” and “applied” mathematics. Looking back to his 1918 work on the continuum, one can here see that he continues to think about ways in which an “atomic” view might illuminate the troubled foundations of mathematics. He looks at sphere and torus with the eyes of a physicist, even as he contemplates space-time with a mathematician’s gaze. As he quotes Galileo, one recognizes his kinship with that archetypal mathematician-philosopher. Weyl recognizes in the basic mathematical concept of isomorphism the essence of what physicists call relativity. Symbols can encompass and unify both concepts: here the axiomatic and the constructive aspects of mathematics converge.
Hilbert remained behind in Germany, retired unhappily in the desolate ruin of Göttingen, in his “tragic years of ever deepening loneliness” after 1933. Though he had been deeply disturbed by Weyl’s temporary adherence to “the Revolution,” Hilbert continued to esteem his “favorite son,” especially after Weyl drew away from Brouwer’s intransigance. For his own part, Weyl said that his motto remained: “True to the spirit of Hilbert.” On the occasion of Hilbert’s seventieth birthday (1932), Weyl wrote: “Woe to the youth that fails to be touched to the core by such a man as Hilbert!”26 Hilbert died in 1944, in the depths of the war. Weyl memorialized him in two essays, the first less detailed and more personal, the second a deep and wide-ranging survey that covers, in broad outline (but with considerable richness of detail), Hilbert’s preoccupations and achievements.
These are complementary accounts and can be read in different ways, according to the reader’s desire to penetrate more or less deeply into the substance of Hilbert’s work. Both are suffused with Weyl’s own distinctive style, eloquent, profound, and more expansive than we are used to in our hurried times. Even when Weyl is describing the content of a mathematical theory, he writes personally and expressively; he feels the direction and import of mathematical ideas no less than the force of human character. Indeed, for Weyl the mathematical and the human converge not merely in the context of anecdote or recollection but in his keen response to the felt impact of both. Weyl registers surprise and wonder, both at human and mathematical beings. As he notes about the deep problems of the foundations of mathematics, “‘mathematizing’ may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.”27
Weyl’s portrait of Hilbert is filled with amazing insights, from his close perspective as student and colleague over many years. Who else could have described “the peculiarly Hilbertian brand of mathematical thinking” as “a swift walk through a sunny open landscape; you look freely around, demarcation lines and connecting roads are pointed out to you, before you must brace yourself to climb the hill; then the path goes straight up, no ambling around, no detours.” The words Weyl uses to describe how Hilbert transformed geometry eloquently describes his own amazing accomplishment in this essay: through his consummate artistry and insight, we see Weyl, no less than Hilbert, “as if one looked into a face thoroughly familiar and yet sublimely transfigured.” This comes about through Weyl’s familiarity with every side of Hilbert’s work — indeed, with every side of mathematics — which he seems to have lived through and felt, not just mastered.28
Weyl’s amazing use of words deserves special attention, and not merely because of the great beauty of what he says. He understands that, even in the seemingly trans-linguistic realm of mathematics, words are essential tools, especially in light of Hilbert’s radical reconsideration of its formalism as a kind of game. As Weyl notes, “already in communicating the rules of the game we must count on understanding. The game is played in silence, but the rules must be told and any reasoning about it, in particular about its consistency, communicated by words.”29 In his discussion of Hilbert’s new axiomatics, Weyl thus deepens the connection between his own symbolic view — within which words play a central role — and Hilbert’s formalism.
Weyl also emphasizes the breadth of Hilbert’s accomplishment; though Hilbert is too consistent and forthright to be a shape-shifter, the many-sided Weyl is specially equipped to speak about his teacher’s manifold contributions across mathematics. In each section, Weyl himself seems transformed into the ideal guide in each different section, precise and incisive in his discussions of algebra, profoundly discursive about foundational issues, steeped in Fredholm’s theory when integral equations come to the fore. Even the preoccupation with physics is shared between Hilbert and Weyl, showing another aspect of their temperamental similarity.
“A tree is best measured when it is down,” the saying goes, but when in 1946a Weyl wrote his preface to a review [1946b] of a volume of essays about Bertrand Russell, his subject was still very much alive and active as a public figure. Weyl takes this occasion to give a wonderful overview of the realms of modern logic, to which Russell had contributed so signally.30 Russell’s own lucidity is equalled by the clarity with which Weyl recounts his ideas, including touches of wry humor (such as comparing Russell’s theory of types to the dogmas of the Church Fathers, as little as Russell could be accused of religious dogmatism or indeed of any kind of conventional religiosity). Such humor aside, the cosmopolitan Weyl acclaims his no-less-worldly friend for the “incomparable riches” of U, the “Russell universe.” Russell’s scheme of setting up a hierarchy of types in order to avoid antinomies and paradoxes (such as Russell himself had pointed out) might be compared with the levels of infinity that Weyl himself had considered. Still, Weyl is quite clear about what he considers the limitations of the project of Russell and Whitehead’s Principia Mathematica. Throughout the essay, Weyl seeks to find ways to reconcile or at least confront the axiomatic approach Russell pioneered with the constructive approach Weyl shared with the intuitionists. Weyl ends with a half-humorous diagram that locates his own “universe” W on a diagram, quite a bit to the left of Russell’s U and Ernst Zermelo’s Z, and hovering above Brouwer’s B and Hilbert’s H.31
On March 19, 1949, Weyl joined in the celebration of Einstein’s seventieth birthday; a photograph of that occasion shows a somewhat bemused Einstein surrounded by colleagues, Weyl and Gödel standing close by.32 Weyl’s lecture on “Relativity Theory as a Stimulus in Mathematical Research” reflects his long and deep involvement with Einstein’s ideas, which “made an epoch in my own scientific life.” Weyl emphasizes the long mathematical search for invariants, of which relativity was one facet, along with general geometric invariants and (more surprisingly) even the Galois theory of algebraic solvability. Indeed, Einstein himself had come to realize that the name “invariant theory” would have been a far more accurate choice than “relativity theory,” with its unfortunate associations of indiscriminate relativism and “anything goes.” Weyl gives considerable attention to the work of Élie Cartan, with whom Einstein had an extended correspondence and who, along with Weyl, had demonstrated the naturalness of general relativity as a generalization of Newtonian mechanics. With characteristic modesty and irony Weyl describes himself as “a lone wolf in Zurich” who “was all too prone to mix up his mathematics with physical and philosophical speculations.”33
In 1951, Weyl wrote his magisterial survey of “A Half-Century of Mathematics,” an overview for which he was uniquely qualified, having participated in so many different facets of its development during the five decades of his own mathematical life. In a special way, this is a kind of intellectual autobiography, as if Proteus were to reprise his multiform career, blending his many shapes with the larger stream of mathematical imagination in which he had been so deeply immersed, which “has the inhuman quality of starlight, brilliant and sharp, but cold.” Weyl looks back to the beginnings of his work (and of our anthology) when he begins with mathematics as “the science of the infinite,” but now, decades later, his narration has acquired many more layers of meaning and levels of nuance. For instance, he emphasizes that certain concepts like set and mapping are really premathematical, even though they often appear in texts as if they were simply part of the fabric of mathematics itself. There are also practical, witty touches: he explains the order of inverse transformations T−1S−1 by comparison to the order in which one dresses and undresses, ST, where S = shirt and T = jacket (in that more formally attired day). Weyl constantly connects his exposition with references to the physics of music, light, thermodynamics, and quantum theory. Even in the midst of his discussion of “pure” mathematical fundamentals, he notes that “the general problem of relativity consists in nothing else but to find the group of automorphisms.” He vividly compares the “wildness” or “tameness” of different groups and their representations (a field to which he himself made signal contributions that he, characteristically, does not mention). Without mentioning his own name, Weyl also seems to bid farewell to his own dreams of a unified field theory: “it is probably unsound to try to ‘geometrize’ all physical entities.”34
Weyl returned to many of these themes and placed them in an even larger context in his 1953 essay on “Axiomatic Versus Constructive Procedures in Mathematics,” here published for the first time in its complete, original form. He is especially troubled that “our mathematics of the last decades has wallowed in generalizations and formalizations,” which he explains as a search for simplicity. Despite the general tendency to dilute the “good nourishing soup” of mathematics with “cheap generalizations,” Weyl still praises “the basic soundness and importance of the axiomatic approach.” In this essay, he brings into the foreground a theme that had run through many of his earlier writings, the ultimate complementarity of the axiomatic and constructive approaches. Though the desire to do justice to axiomatics underlies his even-handed presentation, ultimately he confesses his own long-standing predilection, that “my own heart draws me to the side of contructivism.”35
This anthology ends with what may well be Weyl’s last essay, which has never before appeared in print. He delivered “Why is the World Four-Dimensional?” as a lecture in Washington, D.C. on March 29, 1955, only nine months before his death of a sudden heart attack soon after his seventieth birthday.36 After retiring from the Institute for Advanced Study in 1951, Weyl had been going back and forth between America and Switzerland. Returning to the site of his youthful freedom, Weyl revisited some of the thoughts that had germinated there. During the course of his intensive work on general relativity in 1918, he had begun to think about what deeper mathematical and physical reasons might underlie the four-dimensionality of space-time. In the course of introducing his arguments about gauge invariance, he also confronted the question of conformal invariance, which arbitrarily rescales the fundamental unit of length. Weyl was also versed in the conformal mappings of complex variables, which he explored in his pioneering book The Idea of a Riemann Surface (1913); here again his protean perspective enabled him to make connections between seemingly diverse fields. From these considerations, in 1919 Weyl had suggested that the mathematical structure of Maxwell’s equations necessitated three dimensions of space and one of time as related in the four-dimensional manifold introduced by his teacher Hermann Minkowski (1908). As usual, Weyl presents his conclusion with unforgettable style:
Suppose that a single light, a candle, is burning in the world. Now blow this candle out; what will happen according to the Maxwellian laws? You probably think it will grow dark, pitch dark, in a sphere around the candle which expands with the velocity of light. And you are right — provided the number n is even, especially in our world for which n = 4. But it would not be so in a world of odd dimensionality. Now here you have a physically interesting difference at least between even and odd dimensions, although it does not single out the dimensionality 4. “And God said, Let there be light: and there was light,” so tells the story of Creation in Genesis, Chapter I. If He wished to keep the possibility open for Himself to say “Let there be darkness again” and to accomplish this by blowing out His candles then He had to make the world of even dimensionality.37
Weyl’s arguments continue a long stream of preceding speculations about dimensionality, reaching back to Aristotle, Galileo, and Kant.38 In this 1955 essay, Weyl reprised his youthful insights, noting the limitations of his early arguments based on Maxwell’s equations but also adding new and suggestive topological arguments. In the twenty-first century, these issues have become increasingly important in the context of string and other theories using higher-dimensional manifolds that need to be connected with four-dimensional space-time. For all these initiatives and for the burgeoning study of conformal field theories, Weyl was, yet again, the pioneer who first gazed on the promised land.39
Aside from the two essays translated from German, the other texts follow Weyl’s printed English version or manuscript without change in punctuation; though his practice does not follow presently standard usage, it seemed better to allow the reader access to his original text. Foreign language terms have been italicized according to current practice, but otherwise Weyl’s original use of italics has been preserved. Notes follow each essay, clarifying the sources of the texts and offering some comments and references, including Weyl’s own. I am responsible for all material in square brackets [···]; though Weyl’s original citations have been kept as he had them, other editorial references are keyed to those given in full at the end of the book. I thank John Grafton for his interest and support of this and so many other books we have done together. I am deeply grateful to Brandon Fogel, Jeremy Gray, Paolo Mancosu, Erhard Scholtz, Thomas Ryckman, and Norman Sieroka for their friendly advice; Philip Bartok, Brandon Fogel, and Norman Sieroka were most generous and helpful in revising the German translations, as was Abe Shenitzer who kindly allowed the reprint here of his translation of one of the essays. This book owes a great deal to the superb technical and editorial collaboration of Alexei Pesic, whom I particularly thank. Finally, I sincerely thank Hermann Weyl’s grandchildren Annemarie Weyl Carr, Peter Weyl, and Thomas Weyl for their gracious interest and their permission to publish these works. Weyl’s sons Michael and Joachim, each extraordinary in his own right, have passed on; I remember with gratitude lively conversations with Michael and dedicate this anthology to their memories.
Raoul Bott recalled that “Weyl’s seminar lectures were not particularly easy to understand, but they always had about them this air of ease, of a natural motion in the inevitable stream of the subject itself. . . . At a crucial point of a lecture, Hermann Weyl had the habit of lifting his shoulders and then letting them fall again. This motion seemed to convey the inevitability of that particular turn of thought, the God-given nature of our subject, and the minor role that he himself might have had in its development.” Looking back on the whole arc of Weyl’s career, Dyson judged that “so long as he was alive, he embodied a living contact between the main lines of advance in pure mathematics and in theoretical physics. Now he is dead, the contact is broken, and our hopes of comprehending the physical universe by a direct use of creative mathematical imagination are for the time being ended.”40 May this collection help bring Weyl’s protean imagination back to life for a new generation.
PETER PESIC
1 Chevalley and Weil 1957;WGA 4:655; Dyson 1956, 457; Newman 1957, 308.
2 See below, 95
3 See Weyl 1953 for a valuable account of the German university system especially as he had known it before the Second World War: “the Privatdozent, unlike the professor, is . . . not an appointee of the state. . .. He has the right to lecture but no obligations whatsoever. Therefore, he can devote his whole time and energy to research and to giving lectures . . . on such topics as are of interest to him.” For Weyl and the ETH, see Frei and Stammbach 1992.
4 Weyl’s recollections come from 2009a, 168; the Einstein quotes from Einstein 1987–, 8:669–670 [491]; 8:710 [522] (emphasis original). Regarding the discussions between Einstein and Weyl about his 1918 theory, see Ehlers 1988, Scholz 1994, 1995, 1999a, 2001b, 2004, 2009 Sigursson 1991, Fogel 2008. On the history of gauge theory, see Scholz 1980, 1994, 1999a, 1999b, 2001b, 2004, O’Raifeartaigh 1997, 2000, Ryckman 1996, 2003, 2007, Straumann 2001.
5 It should be noted that there is considerable overlap between this essay and Weyl’s lecture on “Infinity” from The Open World, Weyl 2009a, 66–82.
6 Ewald 1996, 2:838–940 contains an excellent selection of Cantor’s writings and correspondence.
7 For Kronecker, see Ewald 1996, 2:941–955, which discusses the source of his famous aphorism on 942; see also Weyl’s comment on 94, below, and Edwards 1995.
8 For helpful overviews of the intuitionistic controversy (and Weyl’s place in it), see Kleene 1967, 191–201; van Dalen 1999, 291–301, 307–312, 316–326, 375–376, 390–391; Bell 2000; Hesseling 2003; and the essays in Shapiro 2005, 356–411. For a selection of Brouwer’s writings, see Benacerraf and Putnam 1964, 66–84; Ewald 1996, 2:1166–1207; Mancosu 1998, 1–64; and Gray 2008, 290–304, 413–424, which admirably contextualizes mathematics in the larger currents of modernity.
9 Other arguments have shown that 
is in fact irrational, but they are difficult and no easy proof is known. See Gel’fond 1960.
10 Wittgenstein 1975, §§ 145, 173–174. For his reaction to Brouwer, see Hesseling 2003, 190–198, who argues that it was “the foundational battle between Brouwer and Hilbert which stimulated Wittgenstein to hold his later views.” (197) For Wittgenstein’s reactions to Weyl’s work, see Hesseling 2003, 195.
11 For Weyl’s work on gauge theories, see Weyl 2009a, 3-5, Ryckman 2003, Fogel 2008, 45–120.
12 See Weyl 1994 and Scholz 1994, 1995, 2000; for a contemporary reconsideration (and even vindication) of this work, see Feferman 1998, 51–58, 249–283; Feferman 2000, and his article in Shapiro 2005, 590–624.
13 For a helpful selections of Hilbert’s writings, see Benacerraf and Putnam 1964, 134–151; Ewald 1996, 2: 1087–1165; and Mancosu 1998, 149–274.
14 See below, 17–19.
15 See below, 21. Weyl refers, in a footnote below (83), to Ernst Cassirer, whose magisterial work on symbolic forms [1953-1996] are comparable to Weyl’s preoccupations; for his critical commentary, see Weyl 2009b, 149–150, 216. see also Weyl 2009a, 194–195. For discussion of Weyl’s symbolic constructivism, see Scholz 2006a, 2011b, and Toader 2011.
16 See below, 28, 30. For Weyl’s relation to Husserl, see Van Dalen 1984; Tonietti 1988; Feist 2002; Hesseling 2003, 106–107, 124–132; Tieszen 2005, 248– 275; Ryckman 2007; Sieroka 2007; Gray 2008, 204–210; Mancosu 2011, 259–345; and Toader 2011. Meister Eckhart was an early fourteenth century mystical theologian who distinguished between God and the godhead, as a source of freely overflowing creation. Eckhart’s teachings were accused of heresy by the medieval Church.
17 For examples of Klein’s pedagogical writings, see Klein 1939 and Pesic 2007, 109–116; regarding Klein’s pedagogical work, see Tobies 1981. For Weyl’s 1930 eulogy of Klein, see WGA 3:292–299.
18 See below, 33, 47.
19 For the fight between angel and devil, see Weyl 1939b, 681; for the Göttingen community, see Sigurdsson 1994, 1996, 2007; for Weyl’s relations with Gödel, see Dawson 1997, 202.
20 Einstein’s comment was in a letter to the New York Times on 3 May 1935, included (with Weyl’s address) in Catt 1935 and Dick 1981, 92–94; for Weyl’s quote, see below 65.
21 Reid 1996, 143.
22 See below, 56. Roquette 2008 gives a helpful picture of the relations between Weyl and Noether. For Noether’s influence, see McLarty 2006.
23 Weyl also gave special courses of lectures at Yale (The Open World, 1932) and Swarthmore (Mind and Nature, 1934) and addressed the Princeton (1946) and Columbia (1954) bicentennial conferences; these are all included in Weyl 2009a.
24 See below, 67.
25 See below, 69, 70, 72.
26 Reid 1996, 200–201.
27 See below, 88.
28 See below, 97, 113.
29 See below, 117.
30 Because Weyl’s actual review [1946b] discusses in detail the various articles in a volume devoted to Russell’s work [Schilpp 1951], it seemed preferable only to include here Weyl’s self-sufficient preface [1946a].
31 For this diagram, see below, 144.
32 For this photo, see Weyl 2009a, 159.
33 Weyl 2009a, 168 records his feelings about the role of relativity theory in his life; for the Weyl-Cartan arguments, see Pesic and Boughn 2003. See also the correspondence in Cartan and Einstein 1979; for Weyl’s quote about himself, see below, 152–153.
34 See below, 159, 170–172, 174, 188.
35 See below 196–197, 202.
36 This lecture is listed last on the chronological inventory in the ETH Archiv of the manuscripts in the Weyl Nachlaβ. To the best of my knowledge, no other writing of his later than this lecture has come to light. I thank Christine Di Bella, Archivist at the Shelby White and Leon Levy Archives Center at the Institute for Advanced Study, Princeton, for her invaluable help in searching their records.
37 Quoted from 211, below. Weyl first stated his speculations on Maxwell’s equations limiting the dimensionality of space-time in the third (1919) edition of Space-Time-Matter (Weyl 1952a, 284); his treatment of Riemann surfaces is Weyl 2009c. For the history of conformal invariance, see Kastrup 2008. For the earliest statements about the conformal invariance of Maxwell’s equations, see Bateman 1909, 1910a, 1910b, and Cunningham 1910, though I have found no evidence that Weyl knew these papers.
38 For the history of speculations about dimensionality, see Whitrow 1955 and Barrow 1983. Ehrenfest 1918 independently argued for the three-dimensionality of space based on the stability of dynamical orbits, which he connected with Weyl’s work in his 1920 paper. Weyl does not mention Ehrenfest’s work, though he may well have been aware of it via Einstein, a mutual friend.
39 Many aspects of this field are in rapid development and change; see, for instance, Lämmerzahl and Macias 1993, Tegmark 1997, Nelson and Sakellariadou 2009.
40 Bott 1988; Dyson 1956, 457. For views of Weyl’s mathematical legacy, see Wells 1988 and Tent 2008.