The Mathematical Imagination: On the Origins and Promise of Critical Theory by Matthew Handelman (review)

Any reader interested in the Frankfurt School and the history of Critical Theory who subscribes to the longstanding notion that this school of thought is fundamentally incompatible with quantitative thinking and the field of mathematics is in for a surprise with The Mathematical Imagination. Handelman rejects the notion, articulated with brutal clarity by Max Horkheimer and Theodor Adorno, that the “mathematization of thought typifie[s] the return of enlightenment to barbarism” and shows instead that there is a strong tradition of German-Jewish thinkers who, in the early twentieth century, saw in mathematics a way of grappling with the crises of modernity (8). Although the writers whose work Handelman draws upon to make his case—Gershom Scholem, Franz Rosenzweig, and Siegfried Kracauer—are, in their own right, very idiosyncratic figures, Handelman defines a common element within their intellectual projects that he dubs “negative mathematics.” Although the arguments of the individual chapters are too complex—and perhaps a touch too technical, at times—for easy summary, Handelman’s level of engagement with mathematical theory in its connection to aesthetics is unrivalled among current publications. More or less recent works which touch upon related topics, such as Lynn Gamwell’s Mathematics and Art: a Cultural History (2016) or Max Benzner’s The Sociology of Theodor Adorno (2011), could have benefitted from The Mathematical Imagination in order to add more historical context to comments made in passing about Adorno’s and Horkheimer’s approach to mathematics and the natural sciences.

The foundational concept of The Mathematical Imagination—“negative mathematics”—is suggestive, and also ripe for potential misunderstandings. For Handel-man, it has nothing to do with negative numbers per se, but is instead grounded within an analysis of “mathematical approaches to issues of absence, lack, privation, division, and discontinuity” (10). These approaches take various guises in the case studies of The Mathematical Imagination, but they each have to do with how “mathematics develops concepts and symbols to address ideas that, in some accounts, human cognition and language cannot properly grasp or represent in full, such as the concept of the infinite or even the nature of mathematical objects themselves” (10). After a chapter devoted to re-evaluating an “overdetermined equation of mathematics with instrumental reason” (49) that emerged within debates between the Frankfurt School and the Vienna Circle in the 1930s, Handelman devotes the rest of The Mathematical Imagination to examples where mathematics is used as a “powerful mode of aesthetic signification and cultural analysis” (49). The second chapter discusses how Gershom Scholem’s “negative aesthetics” takes as its point of departure an awareness of the restrictive nature of formal mathematical notation: whereas “for Scholem, a pure, complete mathematical logic would consist of a language of non-semantic, non-phonetic, self-referential signs,” mathematics “restricts representation in language by excluding not only rhetorical symbols [ . . . ] but also the semantically meaningful sounds of human language” (91). What Scholem does, according to Handelman, is to recodify this foundational lack in terms of formal literary strategies and an aesthetic sensibility. In literary terms, this corresponds to the genre of the lament, and Handelman’s original idea is to show how Scholem’s translation of the “Book of Lamentations” and work on the lament are, in fact, his contribution to negative mathematics. “Through the metaphorics of structure defined by the restriction of representation,” he writes, “the theorization of lament as a poetic genre and translations of the biblical lamentations into German transformed the philosophy of mathematics’ approach to negativity into a creative literary strategy” (88).

The two other case studies, on Rosenzweig and Kracauer, take up the problem of a “negative mathematics” from two very different directions. Rosenzweig, Handelman argues, arrives at a negative mathematics via the tools of infinitesimal calculus which “allowed him to grasp what other forms of knowledge and language could not express” (105). The “synthesis of infinitude and finitude” in...