It is often claimed that analysis is grounded on the principle of contradiction alone and synthesis is grounded not only on the principle of contradiction but on pure intuition as well. This distinction is inaccurate. In my talk, I discuss the notion of analysis as something that can be grounded on sensibility as well. For this purpose, I present practices of Greek geometrical analysis and discuss how they shaped philosophical and mathematical notions of analysis that are broader than merely logical analysis. I present the case of the philosopher Salomon Maimon (1753-1800) and his work on the different notions of analysis. Maimon's work on analysis is entwined with his work on invention. When writing the outlines of a theory of invention, he turns to Euclidean geometry and practices of Greek geometrical analysis as his main source of influence. This influence is extended not only to his formation of methods of invention but also to his notions of analysis and invention. He presents several notions of analysis, philosophical and mathematical, that are grounded not only on the principle of contradiction but on intuition as well. My discussion of such influences will be accompanied by examples taken from Euclid's Elements and Data. This study of the different forms of analysis is meant to shed light on the less known aspects of the concept and its practices.

As is well known in the history of mathematics, the path to the invention of calculus in late seventeenth-century Europe passed through Buonaventura Cavalieri’s geometry of “indivisibles,” the infinitesimally small slices into which he proposed dividing geometric figures in order to compute the total area contained within their boundaries. The ontological status of these indivisibles was, however, a vexed issue, and the problem of how to deal with the infinitely small would remain a source of much contention for centuries -- as is suggested by Bishop Berkeley’s withering description of Newtonian “fluxions” as the “ghosts of departed quantities.” Tracing the path from Cavalieri’s indivisibles through Leibniz’s infinitesimals, my paper will suggest that early modern attempts to render calculable the minutiae of space and motion have a wide cultural resonance, one that becomes especially visible in literary and metaphysical experimentations with sequences and progressions, in such diverse writers as Gaspara da Stampa, Shakespeare, and Milton.

The aim of this paper is to shed light on an understudied aspect of Giordano Bruno's intellectual biography, namely, his career as a mathematical practitioner. Early interpreters, especially, have criticized Bruno's mathematics for being “outdated” or too “concrete”. However, thanks to developments in the study of early modern mathematics and the rediscovery of Bruno's first mathematical writings (four dialogues on Fabrizio's Mordente proportional compass), we are in a position to better understand Bruno's mathematics. In particular, this paper aims to reopen the question of whether Bruno anticipated the concept of infinitesimal quantity. It does so by providing an analysis of the dialogues on Mordente's compass and of the historical circumstances under which those dialogues were written. Mordente's compass was almost unknown until the late 1800s, as its existence was overshadowed by that of another proportional compass, invented by a better-known Italian scientist: Galileo Galilei. However, Mordente's compass did not go completely unnoticed by his contemporaries, catching the eye of technicians and mathematical practitioners, but also of speculative thinkers like Bruno. Puzzled by the novelty of Mordente’s invention, Bruno offered to write an exposition of the compass in the form of dialogues. In these dialogues, in an attempt to provide a theoretical explanation for the use of the compass, Bruno presented the first version of his atomist geometry based on the concept of the "minimum". This minimum was in essence an infinitely small quantity. As such, I argue that it can be regarded as a forerunner of the infinitesimals.

This paper scrutinizes the revisions of mathematician Bernhard Riemann’s (1822-1862) 1854 habilitation lecture at the University of Göttingen. It argues that the lecture is a reflection of how mathematicians developed non-Euclidean geometries in the nineteenth century, breaking with long-standing professional conventions and philosophical convictions in order to do so. Riemann’s concept of the “manifold,” which he presented in this lecture, was one of the most widespread non-Euclidean frameworks in the nineteenth century, and endures as a foundational concept in mathematics today. This paper argues that, while the manifold (and non-Euclidean geometry) was a “rupture,” it was also continuous with the mathematical practices that came before it, including the study of minimal surfaces. More broadly, Riemann’s papers reveal surprising aspects of mathematical practice, at exactly the moment when mathematics purportedly became abstract, immaterial, and unempirical. Riemann’s mathematical research directly addressed questions of religion and metaphysics: he argued that the “world manifold” was the mechanism connecting human souls to the “world soul.” And Riemann described mathematics as though it could act, and act against him; he frequently was so captivated by his research that he could not pull himself away until he became physically ill. By using Riemann’s revisions to temporally reconstruct the creation of the manifold, this paper challenges two narratives, one historiographical and one cultural: the myth of non-Euclidean geometry as a total rupture, and the notion that mathematics is immaterial and disembodied.