Question

The standard historical narrative surrounding the development of mathematical analysis in the nineteenth century characterizes the era as one defined by the "arithmetization of analysis." In this view, mathematicians such as Karl Weierstrass and Augustin-Louis Cauchy recognized the inherent logical vulnerabilities of eighteenth-century calculus, which relied heavily on spatial metaphors, geometric intuition, and the philosophically dubious manipulation of infinitesimals—quantities treated paradoxically as both zero and non-zero. To secure the foundations of mathematics, these pioneers supposedly launched an epistemic crusade to purge analysis of geometry and kinematics, replacing intuitive heuristics with strictly arithmetic and logical formulations, most notably the epsilon-delta definition of the limit. According to this traditional historiography, the drive for rigor was an exclusive product of the internal intellectual evolution of mathematics—a teleological march toward absolute logical certainty.

However, recent scholarship by historians of mathematics has begun to fracture this monolithic narrative by highlighting the profound institutional and sociological pressures that catalyzed the rigorization process. As scholars like Gert Schubring have documented, the nineteenth century witnessed the unprecedented institutionalization of mathematics as a core discipline within the rapidly expanding Prussian university system. This institutional shift necessitated the pedagogical transformation of mathematics from a niche pursuit of royal academies into a standardized curriculum suitable for mass instruction. The idiosyncratic, intuition-heavy methodologies of eighteenth-century mathematicians—which often relied on the unstated geometric assumptions and personal ingenuity of the individual practitioner—were inherently resistant to standardization. University professors found that they could not reliably teach large lecture halls of students using methods that demanded an innate spatial intuition rather than a reproducible logical sequence.

Consequently, the demand for rigorous, step-by-step arithmetic proofs emerged largely as a pedagogical necessity rather than a purely philosophical one. Weierstrass’s meticulous definitions provided a sequential, logically transparent framework that could be uniformly transmitted to students, independent of their varying capacities for geometric visualization. Furthermore, historians note that the supposed total eradication of geometry from analysis is a retroactive exaggeration; archival evidence of Weierstrass's own lecture notes reveals that he frequently employed spatial diagrams as pedagogical scaffolds, even as he formally eschewed them in his published proofs. Thus, the arithmetization of analysis was not merely the triumph of pure logic over flawed intuition, but a pragmatic adaptation to the sociological demands of an emerging educational infrastructure.

According to the passage, which of the following is explicitly cited by modern historians as a reason why eighteenth-century mathematical methodologies were inadequate for nineteenth-century Prussian universities?