The Veil of Maya
The attraction of everything problematic, the joy of X, however, is too great in such more spiritual, more spiritualized people, for this joy not to flare up again and again like a bright blaze over all the distress of what is problematic, over all the danger of uncertainty, even over the jealousy of the lover. We know a new happiness.1
Scholarship seems not to have considered which metaphorical veil, or Schleier, Hilbert was inviting his listeners to lift, among the many on offer to an educated German in the late nineteenth century. Was it the “veil of poetry” [der Dichtung Schleier] a radiant goddess handed to Goethe, revealing herself as Truth; or was it one of the seven veils of Oscar Wilde’s Salomé, which Richard Strauss orchestrated, a few years after Hilbert’s lecture, in the Schleiertanz that concludes his opera?
Rather than assuming Hilbert sought to awaken the prurient inclinations of coming generations, it is safest to assume that he was thinking of of the “veil of Isis” that Schiller introduced into German poetry, that Kant identified with the barrier separating human understanding from the Ding an sich, that Schlegel enjoined his reader to tear—“Whoever cannot bear the view of the goddess, let him flee or perish.” The piercing of this veil has enjoyed a long career as a metaphor for the pursuit of the scientific knowledge of nature, not least in the title—Isis—of what is still one of the most prestigious international journals of history of science.
Hilbert’s listeners in Paris knew that lifting veils is a risky business. In Schiller’s poem The Veiled Statue at Saïs [Das verschleierte Bild zu Sais], a young seeker of Truth learns that it is hidden behind the veil of the goddess. Ignoring a priest’s warning, he raises Isis’s veil and falls “senseless and pale.” “For all time, the serenity had gone from his life. A deep melancholy carried him off to an early grave.” Schlegel’s comment was a direct retort to Schiller’s poem, and Novalis agreed: “If it is true that no mortal can lift the veil … then we will just have to try to become immortal.”2
Hilbert spoke in Paris in the spirit of Novalis. He famously affirmed his turn-of-the-century optimism in another sentence from the same lecture: “Here is the problem, seek the solution. You can find it through pure thought, because in mathematics there is no Ignorabimus.”* Mehrtens sees Hilbert’s near-contemporary Felix Hausdorff as a partisan of a different strand of modernism, one less confident that mathematics was destined to unveil an unequivocal Truth. Writing in a literary journal two years after Hilbert’s lecture, under the pseudonym Paul Mongré, Hausdorff imagined a Truth-seeker in his garret, staring at his navel and thinking “Everything is thus, but must it be thus? … Could there not be a crystal space, where one could see around the corner and sense one’s way into another I [in ein anderes Ich hineinempfinden]? … Or why not three genders … Men, Middlers, Mothers….” Mongré’s essay—his-
torians translate the pseudonym as “Paul to-my-liking”—is entitled “The Veil of Maya” [Der Schleier der Maja],3 unquestionably a reference to Schopenhauer, by way of Nietzsche, philosophers traditionally seen as more sympathetic to the arts than to the sciences.
Classic accounts of the modernist turn in mathematics frame it as a confrontation between camps led by Hilbert, the optimist formalist, and by L.E.J. Brouwer, founder of intuitionism and critic of Cantor’s abstract set theory. Mehrtens brackets Hausdorff between Hilbert and Brouwer—a modernist (like Hilbert) and “working mathematician,” who devoted considerable attention to philosophy (like Brouwer). Hilbert’s is by far the biggest name in the word cloud of figure 2.2, and Brouwer’s looms large beside his, but Hausdorff’s name is absent, which doesn’t seem fair, because it was he as much as anyone who made Cantor’s set theory speak the language of geometry—not the Euclidean geometry of circles, triangles, and straight lines, but the geometry of our intuition of continuous space.4 It was Hausdorff who invented the term topological space, now indispensable, for the spaces of his geometry. The spaces Hausdorff called “topological” are now known as Hausdorff spaces; they fit marvelously with our intuitions of geometry freed of notions of length, but more flexible axioms have also proved useful, notably in Grothendieck’s geometry. A direct line, or ladder, can be drawn from Hausdorff’s conception of space to Grothendieck’s, passing through Brouwer and Weil, among others, and on past Grothendieck to ∞-categories, as well as the claim I imagined I heard Beilinson make, that geometry is an illusion, a Veil of Maya, that serves to hold fixed points together.
The (small-f) foundations of the topology studied today were largely established by Hausdorff, together with contemporaries in France and among the students of the name worshippers Luzin and Egorov in Moscow. Hausdorff lends his name to the mnemonic math majors use to distinguish the spaces he defined among all topological spaces: “a Hausdorff space is one in which any two points can be housed off from one another.” This is the kind of sophomoric humor that would appeal to the author of Against the Day’s mathematical jokes, and it’s possible Hausdorff himself would have appreciated it. A contemporary reviewer of his topological masterpiece, the Grundzüge der Mengenlehre, published in 1914, noted its “occasional glimmer of humor.” Paul Mongré allowed his humor much freer reign; his greatest literary success, a one-act satirical comedy, was performed more than three hundred times throughout the German-speaking world in the years leading up to World War I.5
Mongré’s principal philosophical text, published when Hausdorff was thirty, was called Das Chaos in kosmischer Auslese [Chaos in cosmic selection]. It “aimed at nothing less than to destroy permanently every type of metaphysics.” Mongré called this position “transcendent nihilism.” The argument, derived from a “kind of metaphorical use of [set-theoretic] mathematics,” was summarized by a contemporary reviewer: “it is impossible to attribute to the system of limitations and syntheses which define our reality any ulterior objective precedence above other systems, apart from its simple relation to us.”6 This is the theme of Der Schleier der Maja’s “Truth-seeker,” but Mongré was not dogmatically attached to Truth; in his Sant’ Ilario, he had written that “[i]f not truth itself, then surely the belief in holding truth is, to a dangerous degree, antagonistic to life and murderous for the future [lebensfeindlich und zukunftsmörderisch].”7
Mongré’s transcendent nihilism was complemented by what Hausdorff called “considered empiricism.” Two years before Liebmann was to call mathematics a “free, creative art,” Hausdorff, also giving an inaugural address in Leipzig as an unpaid Professor extraordinarius, spoke of the “creative freedom of [mathematicians’] concept formation [Gedankenbildung],” especially in their treatment of space. Hausdorff’s lecture was devoted to the “Problem of Space” [Raumproblem], and he identified three Spielräume—literally, “play spaces”—in the “ ‘Freedom of Choice’ among hypotheses on the formulation of the mathematical concept of space,”—the Spielraum of thought, of intuition, and of experience [den Spielraum des Denkens, den Spielraum der Anschauung, den Spielraum der Erfahrung].8 It was with respect to the Spielraum of experience as it bears on the study of physical phenomena that Hausdorff saw the need for a considered empiricism.9 The Spielraum of thought, in contrast, was “very large indeed”; it was the source of the “creative freedom” that, Hausdorff added, mathematics had acquired “not without struggle against philosophical attempts at suppression.”10
We encountered that word “freedom” in our very first chapter, in Cantor’s famous dictum that “the essence of mathematics lies in its freedom.” We saw it again in Liebmann’s characterization of mathematics as a “free, creative art” and in Brouwer’s interpretation of real numbers as “free choice sequences”; it can be found in Hilbert’s writings as well.11 In 1942, as a seventy-four-year-old Jew in Nazi Germany, Hausdorff was faced with the prospect of being deported to the Theresienstadt concentration camp from the Endenich cloister in Bonn, where he was about to be interned. Hausdorff, his wife, and his sister-in-law, chose—one would like to say freely, within the very narrow Spielraum still available to them12—to take an overdose of barbiturates and thereby escape the fate that had been ordained for them.
A team of eminent scholars is completing a definitive edition of Hausdorff’s collected works—“unique … in the annals of mathematical publishing”—with the care befitting the literary figure he undoubtedly was. This scholarship has confirmed the immortality of the “Dionysian mathematician”13 in the land of mathematical Giants. A few years ago the prestigious Hausdorff Research Institute for Mathematics was created in Bonn, where Hausdorff spent the greater part of his career, and which, since the war, has been Germany’s principal mathematical center. Here he is honored as, perhaps, the first modern mathematician to give a name to what we have been calling the “relaxed field”—he called it the “Spielraum of thought”—and as a mathematician who never lost his sensitivity to his chosen field’s problematic attractions while remaining fully aware that every veil lifted only reveals another veil.
* “Da ist das Problem, suche die Lösung. Du kannst sie durch reines Denken finden; denn in der Mathematik giebt es, kein Ignorabimus!”