Wozzeck and the Mathematical

1. A Sort of Introduction

According to a myth repeated in various guises for at least the last 2000 years, the fifth-century BC mathematician Hippasus was taken to sea and drowned for spreading knowledge (or better, a gap in knowledge) about what we think of today as "irrational numbers."¹ His fellow Pythagoreans believed that the harmony of nature was determined by the natural numbers, N = {1, 2, 3, … }. Since the line lengths we would now express as (the hypotenuses of right-angled triangles with shorter lengths 1 and 1, 1 and 2, 2 and 3, respectively, according to Pythagoras' theorem), for example, cannot be expressed as the ratio of two natural numbers—i.e., they were literally irrational, the source of the word's current meaning—they were considered "impious."²

The reason the myth has been repeated in many forms down the centuries is that it says something profound about truth. What that something is, though, tends to shift according to the views of the author citing it. It may surprise anyone who only knows the book by reputation to learn that the first chapter of Oswald Spengler's The Decline of the West, vol.I (Der Untergang des Abendlandes, 1918, rev. 1922) is devoted to mathematics. It occupies this leading position for good reason: if Spengler could show that what counted as mathematical truth was the product of culture, then the same would certainly be true of all other, less secure, branches of learning. He put the Hippasus myth at the very beginning of this chapter because, for him, it illustrated: first, that mathematicians themselves disagreed about what counts as truth; and second, how fundamentally different conceptions of mathematics are in "Classical" and "Western" cultures.³

Spengler's work on Decline overlaps with Alban Berg's on Wozzeck (mainly composed 1918–22, first performance 1925) and although the likelihood of direct influence is slim, Berg had absorbed a similar manner of thinking from other contemporaries, such as Karl Kraus and Arnold Schoenberg, which might be broadly characterized as irrationalism or reactionary modernism.⁴ Within this cluster of ideas, Enlightenment rationality, and particularly science, was felt to be causing the disintegration of—using Spengler's terminology—a strong German culture into a weaker civilization.⁵

However, while Wozzeck clearly critiques rationality, Spengler's irrationalism doesn't provide the right framework in which to understand it. Berg was certainly wary of any claim that scientific or technological progress would automatically lead to social progress. At the same time, he was enthusiastic about contemporary science and, quite contrary to Spengler's proto-Foucauldian belief in the cultural construction of scientific knowledge, ingenuously accepted the claims of Wilhelm Fliess—a close friend of Sigmund Freud and important in the development of psychoanalysis—that numbers and numerical patterns shaped the world in fixed and predictable ways.⁶ Because this aspect of Fliess's work has always been rejected as pseudo-science, Berg scholars have tended to lump Berg's interest in Fliess together with his forays into number mysticism, even though Fliess was actually a serious scientist now remembered as one of the originators of circadian rhythms.

Contrary to the dominant idea that turn-of-the-century mysticism was part of a reaction against scientific modernism, many now think that pseudo-science and the occult provided non-scientist intellectuals with access to the latest mathematical and scientific results and, most pertinently here, their surprising and unsettling ontological and epistemological consequences.⁷ The point of this article is to revisit the mathematical patterning in the music for Wozzeck—including the old forms that have provoked so much discussion over the past century—and reinterpret its effect on the opera in terms of the mathematical turn in physics, the development of non-Euclidean geometry, and the failure of Gottlieb Frege to ground mathematics outside the human subject. Berg absorbed these mathematical shocks in degraded and distorted forms from, for example, Wilhelm Fliess's numerology and Rudolf Steiner's ruminations on the fourth dimension, as well as contemporaneous artistic practices, not least Blaue Reiter expressionism.

Taking Berg's use of mathematics seriously...