Landscapes of the Mind: The Spatial Configuration of Mathematics in Hermann Broch’s Die Unbekannte Größe

In Die Unbekannte Größe (1933) by Hermann Broch (1886–1951), the protagonist – mathematics doctoral candidate Richard Hieck – waits for his brother to fall asleep so that he can work in silence. In the subsequent solitude, he has the following vision:

Und als die Atemzüge seines Schlummers leise vernehmlich wurden, da lichtete sich für Richard der Nebel, er sah eine kristallische Landschaft vor sich [. . .] eine erleuchtet sternenhafte Landschaft, in der die Zahlengruppen [. . .] so leicht einzuordnen waren, daß man [sie . . .] in eine beglückend logische und gleichzeitig ein wenig karussellhafte Bewegung versetzen konnte. Und wenn es auch noch nicht die Lösung [. . .] war, die aus den derart bewegten Abwandlungen der Zahlenkonstellationen sichtbar wurde, so vollzog sich damit in dem [. . .] Schädel Richard Hiecks doch etwas, das als Vorstoß in mathematisches Neu-land zu gelten hatte, [. . .] und es wurde ein Stück der komplizierten unendlichen und niemals ausschreitbaren Gleichgewichtskonstruktion bloßgelegt, die an sich aus leeren Beziehungen besteht und trotzdem das Wunderwerk der Mathematik ist.

(22–23)

The landscape that Hieck sees is not the night sky over an immense expanse of land or water. Instead, he sits within the claustrophobic confines of a small, shared bedroom in a crowded city. No literal constellations illuminate the darkness but rather the numbers and the mathematics they constitute. Mathematics embodies for him a physical space, albeit an imagined one: abstract in its content; concrete in its mental manifestation.

In Broch’s novel, Hieck must navigate a number of spaces: the Viennese cityscape of the 1920s, with its bustling street traffic, parks, and public swimming pool; the constrictive and crowded apartment he shares with his mother, a sister, and a brother; the stratified social structure of the university; that portion of the universe accessible by the powerful telescope he uses while working at the planetarium; the seemingly impenetrable world of women and their desires; and, as in the quotation just read, the space mathematics occupies in his mind. In The Production of Space, Henri Lefebvre attempts to establish a unified theory that locates and describes common qualities of form, structure, and function in what he calls physical, social, and mental space, the first referring to the realm of sensory phenomena, whether real, perceived, or imagined; the second to that of human social structures. Mental space, which pertains to the body of human knowledge including formal and logical abstractions, is what is of concern here. Lefebvre, although cautioning against a premature conclusion that social space derives directly from the abstract concepts of physical and mathematical models of space, nevertheless emphasizes that no one “can grasp ‘reality’ – i.e. social and spatial practice – without starting out from a mental space, without proceeding from the abstract to the concrete” (415). Although no doubt extremely complex in its practical differentiation from its abstract origins, social space, like language and other pragmatic manifestations of mental functions, must begin in some form as cognition. Mind definitely precedes matter. For Hieck, however, this relationship seems even more complex than Lefebvre’s in no way simple theory. Perhaps, on the one hand, mind does indeed precede matter; the physical spaces mentioned above, in which Hieck literally moves, could not have come into existence without some previously constructed mental image of them. But on the other hand, Hieck’s conception of the mathematics that occupies his mental space seems, in turn, to reflect spatial resemblance to the physical spaces of the natural and social environments. This study explores to what extent notions of space develop in the opposite direction to Lefebvre’s cause-and-effect formula: to what extent matter precedes mind; to what extent mental concepts begin with an abstraction of social realities. In other words, it will consider how the authors’ texts can illuminate the interrelationships of modern “real” and “ideal” spaces in order to investigate to what extent “each of these two kinds of space involves, underpins and presupposes the other” (Lefebvre 14). By taking a closer look at Broch’s philosophy, the upheaval in the field of mathematics at the beginning of the twentieth century, known...